This is gsl-ref.info, produced by makeinfo version 4.8 from gsl-ref.texi. INFO-DIR-SECTION Scientific software START-INFO-DIR-ENTRY * gsl-ref: (gsl-ref). GNU Scientific Library - Reference END-INFO-DIR-ENTRY Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 The GSL Team. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with the Invariant Sections being "GNU General Public License" and "Free Software Needs Free Documentation", the Front-Cover text being "A GNU Manual", and with the Back-Cover Text being (a) (see below). A copy of the license is included in the section entitled "GNU Free Documentation License". (a) The Back-Cover Text is: "You have freedom to copy and modify this GNU Manual, like GNU software."  File: gsl-ref.info, Node: Top, Next: Introduction, Prev: (dir), Up: (dir) GSL *** This file documents the GNU Scientific Library (GSL), a collection of numerical routines for scientific computing. It corresponds to release 1.9 of the library. Please report any errors in this manual to . More information about GSL can be found at the project homepage, `http://www.gnu.org/software/gsl/'. Printed copies of this manual can be purchased from Network Theory Ltd at `http://www.network-theory.co.uk/gsl/manual/'. The money raised from sales of the manual helps support the development of GSL. A Japanese translation of this manual is available from the GSL project homepage thanks to Daisuke Tominaga. Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 The GSL Team. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with the Invariant Sections being "GNU General Public License" and "Free Software Needs Free Documentation", the Front-Cover text being "A GNU Manual", and with the Back-Cover Text being (a) (see below). A copy of the license is included in the section entitled "GNU Free Documentation License". (a) The Back-Cover Text is: "You have freedom to copy and modify this GNU Manual, like GNU software." * Menu: * Introduction:: * Using the library:: * Error Handling:: * Mathematical Functions:: * Complex Numbers:: * Polynomials:: * Special Functions:: * Vectors and Matrices:: * Permutations:: * Combinations:: * Sorting:: * BLAS Support:: * Linear Algebra:: * Eigensystems:: * Fast Fourier Transforms:: * Numerical Integration:: * Random Number Generation:: * Quasi-Random Sequences:: * Random Number Distributions:: * Statistics:: * Histograms:: * N-tuples:: * Monte Carlo Integration:: * Simulated Annealing:: * Ordinary Differential Equations:: * Interpolation:: * Numerical Differentiation:: * Chebyshev Approximations:: * Series Acceleration:: * Wavelet Transforms:: * Discrete Hankel Transforms:: * One dimensional Root-Finding:: * One dimensional Minimization:: * Multidimensional Root-Finding:: * Multidimensional Minimization:: * Least-Squares Fitting:: * Nonlinear Least-Squares Fitting:: * Basis Splines:: * Physical Constants:: * IEEE floating-point arithmetic:: * Debugging Numerical Programs:: * Contributors to GSL:: * Autoconf Macros:: * GSL CBLAS Library:: * Free Software Needs Free Documentation:: * GNU General Public License:: * GNU Free Documentation License:: * Function Index:: * Variable Index:: * Type Index:: * Concept Index::  File: gsl-ref.info, Node: Introduction, Next: Using the library, Prev: Top, Up: Top 1 Introduction ************** The GNU Scientific Library (GSL) is a collection of routines for numerical computing. The routines have been written from scratch in C, and present a modern Applications Programming Interface (API) for C programmers, allowing wrappers to be written for very high level languages. The source code is distributed under the GNU General Public License. * Menu: * Routines available in GSL:: * GSL is Free Software:: * Obtaining GSL:: * No Warranty:: * Reporting Bugs:: * Further Information:: * Conventions used in this manual::  File: gsl-ref.info, Node: Routines available in GSL, Next: GSL is Free Software, Up: Introduction 1.1 Routines available in GSL ============================= The library covers a wide range of topics in numerical computing. Routines are available for the following areas, Complex Numbers Roots of Polynomials Special Functions Vectors and Matrices Permutations Combinations Sorting BLAS Support Linear Algebra CBLAS Library Fast Fourier Transforms Eigensystems Random Numbers Quadrature Random Distributions Quasi-Random Sequences Histograms Statistics Monte Carlo Integration N-Tuples Differential Equations Simulated Annealing Numerical Differentiation Interpolation Series Acceleration Chebyshev Approximations Root-Finding Discrete Hankel Transforms Least-Squares Fitting Minimization IEEE Floating-Point Physical Constants Wavelets The use of these routines is described in this manual. Each chapter provides detailed definitions of the functions, followed by example programs and references to the articles on which the algorithms are based. Where possible the routines have been based on reliable public-domain packages such as FFTPACK and QUADPACK, which the developers of GSL have reimplemented in C with modern coding conventions.  File: gsl-ref.info, Node: GSL is Free Software, Next: Obtaining GSL, Prev: Routines available in GSL, Up: Introduction 1.2 GSL is Free Software ======================== The subroutines in the GNU Scientific Library are "free software"; this means that everyone is free to use them, and to redistribute them in other free programs. The library is not in the public domain; it is copyrighted and there are conditions on its distribution. These conditions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of the software that they might get from you. Specifically, we want to make sure that you have the right to share copies of programs that you are given which use the GNU Scientific Library, that you receive their source code or else can get it if you want it, that you can change these programs or use pieces of them in new free programs, and that you know you can do these things. To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of any code which uses the GNU Scientific Library, you must give the recipients all the rights that you have received. You must make sure that they, too, receive or can get the source code, both to the library and the code which uses it. And you must tell them their rights. This means that the library should not be redistributed in proprietary programs. Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU Scientific Library. If these programs are modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. The precise conditions for the distribution of software related to the GNU Scientific Library are found in the GNU General Public License (*note GNU General Public License::). Further information about this license is available from the GNU Project webpage `Frequently Asked Questions about the GNU GPL', `http://www.gnu.org/copyleft/gpl-faq.html' The Free Software Foundation also operates a license consulting service for commercial users (contact details available from `http://www.fsf.org/').  File: gsl-ref.info, Node: Obtaining GSL, Next: No Warranty, Prev: GSL is Free Software, Up: Introduction 1.3 Obtaining GSL ================= The source code for the library can be obtained in different ways, by copying it from a friend, purchasing it on CDROM or downloading it from the internet. A list of public ftp servers which carry the source code can be found on the GNU website, `http://www.gnu.org/software/gsl/' The preferred platform for the library is a GNU system, which allows it to take advantage of additional features in the GNU C compiler and GNU C library. However, the library is fully portable and should compile on most systems with a C compiler. Precompiled versions of the library can be purchased from commercial redistributors listed on the website above. Announcements of new releases, updates and other relevant events are made on the `info-gsl@gnu.org' mailing list. To subscribe to this low-volume list, send an email of the following form: To: info-gsl-request@gnu.org Subject: subscribe You will receive a response asking you to reply in order to confirm your subscription.  File: gsl-ref.info, Node: No Warranty, Next: Reporting Bugs, Prev: Obtaining GSL, Up: Introduction 1.4 No Warranty =============== The software described in this manual has no warranty, it is provided "as is". It is your responsibility to validate the behavior of the routines and their accuracy using the source code provided, or to purchase support and warranties from commercial redistributors. Consult the GNU General Public license for further details (*note GNU General Public License::).  File: gsl-ref.info, Node: Reporting Bugs, Next: Further Information, Prev: No Warranty, Up: Introduction 1.5 Reporting Bugs ================== A list of known bugs can be found in the `BUGS' file included in the GSL distribution. Details of compilation problems can be found in the `INSTALL' file. If you find a bug which is not listed in these files, please report it to . All bug reports should include: * The version number of GSL * The hardware and operating system * The compiler used, including version number and compilation options * A description of the bug behavior * A short program which exercises the bug It is useful if you can check whether the same problem occurs when the library is compiled without optimization. Thank you. Any errors or omissions in this manual can also be reported to the same address.  File: gsl-ref.info, Node: Further Information, Next: Conventions used in this manual, Prev: Reporting Bugs, Up: Introduction 1.6 Further Information ======================= Additional information, including online copies of this manual, links to related projects, and mailing list archives are available from the website mentioned above. Any questions about the use and installation of the library can be asked on the mailing list `help-gsl@gnu.org'. To subscribe to this list, send an email of the following form: To: help-gsl-request@gnu.org Subject: subscribe This mailing list can be used to ask questions not covered by this manual, and to contact the developers of the library. If you would like to refer to the GNU Scientific Library in a journal article, the recommended way is to cite this reference manual, e.g. `M. Galassi et al, GNU Scientific Library Reference Manual (2nd Ed.), ISBN 0954161734'. If you want to give a url, use "`http://www.gnu.org/software/gsl/'".  File: gsl-ref.info, Node: Conventions used in this manual, Prev: Further Information, Up: Introduction 1.7 Conventions used in this manual =================================== This manual contains many examples which can be typed at the keyboard. A command entered at the terminal is shown like this, $ command The first character on the line is the terminal prompt, and should not be typed. The dollar sign `$' is used as the standard prompt in this manual, although some systems may use a different character. The examples assume the use of the GNU operating system. There may be minor differences in the output on other systems. The commands for setting environment variables use the Bourne shell syntax of the standard GNU shell (`bash').  File: gsl-ref.info, Node: Using the library, Next: Error Handling, Prev: Introduction, Up: Top 2 Using the library ******************* This chapter describes how to compile programs that use GSL, and introduces its conventions. * Menu: * An Example Program:: * Compiling and Linking:: * Shared Libraries:: * ANSI C Compliance:: * Inline functions:: * Long double:: * Portability functions:: * Alternative optimized functions:: * Support for different numeric types:: * Compatibility with C++:: * Aliasing of arrays:: * Thread-safety:: * Deprecated Functions:: * Code Reuse::  File: gsl-ref.info, Node: An Example Program, Next: Compiling and Linking, Up: Using the library 2.1 An Example Program ====================== The following short program demonstrates the use of the library by computing the value of the Bessel function J_0(x) for x=5, #include #include int main (void) { double x = 5.0; double y = gsl_sf_bessel_J0 (x); printf ("J0(%g) = %.18e\n", x, y); return 0; } The output is shown below, and should be correct to double-precision accuracy, J0(5) = -1.775967713143382920e-01 The steps needed to compile this program are described in the following sections.  File: gsl-ref.info, Node: Compiling and Linking, Next: Shared Libraries, Prev: An Example Program, Up: Using the library 2.2 Compiling and Linking ========================= The library header files are installed in their own `gsl' directory. You should write any preprocessor include statements with a `gsl/' directory prefix thus, #include If the directory is not installed on the standard search path of your compiler you will also need to provide its location to the preprocessor as a command line flag. The default location of the `gsl' directory is `/usr/local/include/gsl'. A typical compilation command for a source file `example.c' with the GNU C compiler `gcc' is, $ gcc -Wall -I/usr/local/include -c example.c This results in an object file `example.o'. The default include path for `gcc' searches `/usr/local/include' automatically so the `-I' option can actually be omitted when GSL is installed in its default location. * Menu: * Linking programs with the library:: * Linking with an alternative BLAS library::  File: gsl-ref.info, Node: Linking programs with the library, Next: Linking with an alternative BLAS library, Up: Compiling and Linking 2.2.1 Linking programs with the library --------------------------------------- The library is installed as a single file, `libgsl.a'. A shared version of the library `libgsl.so' is also installed on systems that support shared libraries. The default location of these files is `/usr/local/lib'. If this directory is not on the standard search path of your linker you will also need to provide its location as a command line flag. To link against the library you need to specify both the main library and a supporting CBLAS library, which provides standard basic linear algebra subroutines. A suitable CBLAS implementation is provided in the library `libgslcblas.a' if your system does not provide one. The following example shows how to link an application with the library, $ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm The default library path for `gcc' searches `/usr/local/lib' automatically so the `-L' option can be omitted when GSL is installed in its default location.  File: gsl-ref.info, Node: Linking with an alternative BLAS library, Prev: Linking programs with the library, Up: Compiling and Linking 2.2.2 Linking with an alternative BLAS library ---------------------------------------------- The following command line shows how you would link the same application with an alternative CBLAS library called `libcblas', $ gcc example.o -lgsl -lcblas -lm For the best performance an optimized platform-specific CBLAS library should be used for `-lcblas'. The library must conform to the CBLAS standard. The ATLAS package provides a portable high-performance BLAS library with a CBLAS interface. It is free software and should be installed for any work requiring fast vector and matrix operations. The following command line will link with the ATLAS library and its CBLAS interface, $ gcc example.o -lgsl -lcblas -latlas -lm For more information see *Note BLAS Support::.  File: gsl-ref.info, Node: Shared Libraries, Next: ANSI C Compliance, Prev: Compiling and Linking, Up: Using the library 2.3 Shared Libraries ==================== To run a program linked with the shared version of the library the operating system must be able to locate the corresponding `.so' file at runtime. If the library cannot be found, the following error will occur: $ ./a.out ./a.out: error while loading shared libraries: libgsl.so.0: cannot open shared object file: No such file or directory To avoid this error, define the shell variable `LD_LIBRARY_PATH' to include the directory where the library is installed. For example, in the Bourne shell (`/bin/sh' or `/bin/bash'), the library search path can be set with the following commands: $ LD_LIBRARY_PATH=/usr/local/lib:$LD_LIBRARY_PATH $ export LD_LIBRARY_PATH $ ./example In the C-shell (`/bin/csh' or `/bin/tcsh') the equivalent command is, % setenv LD_LIBRARY_PATH /usr/local/lib:$LD_LIBRARY_PATH The standard prompt for the C-shell in the example above is the percent character `%', and should not be typed as part of the command. To save retyping these commands each session they should be placed in an individual or system-wide login file. To compile a statically linked version of the program, use the `-static' flag in `gcc', $ gcc -static example.o -lgsl -lgslcblas -lm  File: gsl-ref.info, Node: ANSI C Compliance, Next: Inline functions, Prev: Shared Libraries, Up: Using the library 2.4 ANSI C Compliance ===================== The library is written in ANSI C and is intended to conform to the ANSI C standard (C89). It should be portable to any system with a working ANSI C compiler. The library does not rely on any non-ANSI extensions in the interface it exports to the user. Programs you write using GSL can be ANSI compliant. Extensions which can be used in a way compatible with pure ANSI C are supported, however, via conditional compilation. This allows the library to take advantage of compiler extensions on those platforms which support them. When an ANSI C feature is known to be broken on a particular system the library will exclude any related functions at compile-time. This should make it impossible to link a program that would use these functions and give incorrect results. To avoid namespace conflicts all exported function names and variables have the prefix `gsl_', while exported macros have the prefix `GSL_'.  File: gsl-ref.info, Node: Inline functions, Next: Long double, Prev: ANSI C Compliance, Up: Using the library 2.5 Inline functions ==================== The `inline' keyword is not part of the original ANSI C standard (C89) and the library does not export any inline function definitions by default. However, the library provides optional inline versions of performance-critical functions by conditional compilation. The inline versions of these functions can be included by defining the macro `HAVE_INLINE' when compiling an application, $ gcc -Wall -c -DHAVE_INLINE example.c If you use `autoconf' this macro can be defined automatically. If you do not define the macro `HAVE_INLINE' then the slower non-inlined versions of the functions will be used instead. Note that the actual usage of the inline keyword is `extern inline', which eliminates unnecessary function definitions in GCC. If the form `extern inline' causes problems with other compilers a stricter autoconf test can be used, see *Note Autoconf Macros::.  File: gsl-ref.info, Node: Long double, Next: Portability functions, Prev: Inline functions, Up: Using the library 2.6 Long double =============== The extended numerical type `long double' is part of the ANSI C standard and should be available in every modern compiler. However, the precision of `long double' is platform dependent, and this should be considered when using it. The IEEE standard only specifies the minimum precision of extended precision numbers, while the precision of `double' is the same on all platforms. In some system libraries the `stdio.h' formatted input/output functions `printf' and `scanf' are not implemented correctly for `long double'. Undefined or incorrect results are avoided by testing these functions during the `configure' stage of library compilation and eliminating certain GSL functions which depend on them if necessary. The corresponding line in the `configure' output looks like this, checking whether printf works with long double... no Consequently when `long double' formatted input/output does not work on a given system it should be impossible to link a program which uses GSL functions dependent on this. If it is necessary to work on a system which does not support formatted `long double' input/output then the options are to use binary formats or to convert `long double' results into `double' for reading and writing.  File: gsl-ref.info, Node: Portability functions, Next: Alternative optimized functions, Prev: Long double, Up: Using the library 2.7 Portability functions ========================= To help in writing portable applications GSL provides some implementations of functions that are found in other libraries, such as the BSD math library. You can write your application to use the native versions of these functions, and substitute the GSL versions via a preprocessor macro if they are unavailable on another platform. For example, after determining whether the BSD function `hypot' is available you can include the following macro definitions in a file `config.h' with your application, /* Substitute gsl_hypot for missing system hypot */ #ifndef HAVE_HYPOT #define hypot gsl_hypot #endif The application source files can then use the include command `#include ' to replace each occurrence of `hypot' by `gsl_hypot' when `hypot' is not available. This substitution can be made automatically if you use `autoconf', see *Note Autoconf Macros::. In most circumstances the best strategy is to use the native versions of these functions when available, and fall back to GSL versions otherwise, since this allows your application to take advantage of any platform-specific optimizations in the system library. This is the strategy used within GSL itself.  File: gsl-ref.info, Node: Alternative optimized functions, Next: Support for different numeric types, Prev: Portability functions, Up: Using the library 2.8 Alternative optimized functions =================================== The main implementation of some functions in the library will not be optimal on all architectures. For example, there are several ways to compute a Gaussian random variate and their relative speeds are platform-dependent. In cases like this the library provides alternative implementations of these functions with the same interface. If you write your application using calls to the standard implementation you can select an alternative version later via a preprocessor definition. It is also possible to introduce your own optimized functions this way while retaining portability. The following lines demonstrate the use of a platform-dependent choice of methods for sampling from the Gaussian distribution, #ifdef SPARC #define gsl_ran_gaussian gsl_ran_gaussian_ratio_method #endif #ifdef INTEL #define gsl_ran_gaussian my_gaussian #endif These lines would be placed in the configuration header file `config.h' of the application, which should then be included by all the source files. Note that the alternative implementations will not produce bit-for-bit identical results, and in the case of random number distributions will produce an entirely different stream of random variates.  File: gsl-ref.info, Node: Support for different numeric types, Next: Compatibility with C++, Prev: Alternative optimized functions, Up: Using the library 2.9 Support for different numeric types ======================================= Many functions in the library are defined for different numeric types. This feature is implemented by varying the name of the function with a type-related modifier--a primitive form of C++ templates. The modifier is inserted into the function name after the initial module prefix. The following table shows the function names defined for all the numeric types of an imaginary module `gsl_foo' with function `fn', gsl_foo_fn double gsl_foo_long_double_fn long double gsl_foo_float_fn float gsl_foo_long_fn long gsl_foo_ulong_fn unsigned long gsl_foo_int_fn int gsl_foo_uint_fn unsigned int gsl_foo_short_fn short gsl_foo_ushort_fn unsigned short gsl_foo_char_fn char gsl_foo_uchar_fn unsigned char The normal numeric precision `double' is considered the default and does not require a suffix. For example, the function `gsl_stats_mean' computes the mean of double precision numbers, while the function `gsl_stats_int_mean' computes the mean of integers. A corresponding scheme is used for library defined types, such as `gsl_vector' and `gsl_matrix'. In this case the modifier is appended to the type name. For example, if a module defines a new type-dependent struct or typedef `gsl_foo' it is modified for other types in the following way, gsl_foo double gsl_foo_long_double long double gsl_foo_float float gsl_foo_long long gsl_foo_ulong unsigned long gsl_foo_int int gsl_foo_uint unsigned int gsl_foo_short short gsl_foo_ushort unsigned short gsl_foo_char char gsl_foo_uchar unsigned char When a module contains type-dependent definitions the library provides individual header files for each type. The filenames are modified as shown in the below. For convenience the default header includes the definitions for all the types. To include only the double precision header file, or any other specific type, use its individual filename. #include All types #include double #include long double #include float #include long #include unsigned long #include int #include unsigned int #include short #include unsigned short #include char #include unsigned char  File: gsl-ref.info, Node: Compatibility with C++, Next: Aliasing of arrays, Prev: Support for different numeric types, Up: Using the library 2.10 Compatibility with C++ =========================== The library header files automatically define functions to have `extern "C"' linkage when included in C++ programs. This allows the functions to be called directly from C++. To use C++ exception handling within user-defined functions passed to the library as parameters, the library must be built with the additional `CFLAGS' compilation option `-fexceptions'.  File: gsl-ref.info, Node: Aliasing of arrays, Next: Thread-safety, Prev: Compatibility with C++, Up: Using the library 2.11 Aliasing of arrays ======================= The library assumes that arrays, vectors and matrices passed as modifiable arguments are not aliased and do not overlap with each other. This removes the need for the library to handle overlapping memory regions as a special case, and allows additional optimizations to be used. If overlapping memory regions are passed as modifiable arguments then the results of such functions will be undefined. If the arguments will not be modified (for example, if a function prototype declares them as `const' arguments) then overlapping or aliased memory regions can be safely used.  File: gsl-ref.info, Node: Thread-safety, Next: Deprecated Functions, Prev: Aliasing of arrays, Up: Using the library 2.12 Thread-safety ================== The library can be used in multi-threaded programs. All the functions are thread-safe, in the sense that they do not use static variables. Memory is always associated with objects and not with functions. For functions which use "workspace" objects as temporary storage the workspaces should be allocated on a per-thread basis. For functions which use "table" objects as read-only memory the tables can be used by multiple threads simultaneously. Table arguments are always declared `const' in function prototypes, to indicate that they may be safely accessed by different threads. There are a small number of static global variables which are used to control the overall behavior of the library (e.g. whether to use range-checking, the function to call on fatal error, etc). These variables are set directly by the user, so they should be initialized once at program startup and not modified by different threads.  File: gsl-ref.info, Node: Deprecated Functions, Next: Code Reuse, Prev: Thread-safety, Up: Using the library 2.13 Deprecated Functions ========================= From time to time, it may be necessary for the definitions of some functions to be altered or removed from the library. In these circumstances the functions will first be declared "deprecated" and then removed from subsequent versions of the library. Functions that are deprecated can be disabled in the current release by setting the preprocessor definition `GSL_DISABLE_DEPRECATED'. This allows existing code to be tested for forwards compatibility.  File: gsl-ref.info, Node: Code Reuse, Prev: Deprecated Functions, Up: Using the library 2.14 Code Reuse =============== Where possible the routines in the library have been written to avoid dependencies between modules and files. This should make it possible to extract individual functions for use in your own applications, without needing to have the whole library installed. You may need to define certain macros such as `GSL_ERROR' and remove some `#include' statements in order to compile the files as standalone units. Reuse of the library code in this way is encouraged, subject to the terms of the GNU General Public License.  File: gsl-ref.info, Node: Error Handling, Next: Mathematical Functions, Prev: Using the library, Up: Top 3 Error Handling **************** This chapter describes the way that GSL functions report and handle errors. By examining the status information returned by every function you can determine whether it succeeded or failed, and if it failed you can find out what the precise cause of failure was. You can also define your own error handling functions to modify the default behavior of the library. The functions described in this section are declared in the header file `gsl_errno.h'. * Menu: * Error Reporting:: * Error Codes:: * Error Handlers:: * Using GSL error reporting in your own functions:: * Error Reporting Examples::  File: gsl-ref.info, Node: Error Reporting, Next: Error Codes, Up: Error Handling 3.1 Error Reporting =================== The library follows the thread-safe error reporting conventions of the POSIX Threads library. Functions return a non-zero error code to indicate an error and `0' to indicate success. int status = gsl_function (...) if (status) { /* an error occurred */ ..... /* status value specifies the type of error */ } The routines report an error whenever they cannot perform the task requested of them. For example, a root-finding function would return a non-zero error code if could not converge to the requested accuracy, or exceeded a limit on the number of iterations. Situations like this are a normal occurrence when using any mathematical library and you should check the return status of the functions that you call. Whenever a routine reports an error the return value specifies the type of error. The return value is analogous to the value of the variable `errno' in the C library. The caller can examine the return code and decide what action to take, including ignoring the error if it is not considered serious. In addition to reporting errors by return codes the library also has an error handler function `gsl_error'. This function is called by other library functions when they report an error, just before they return to the caller. The default behavior of the error handler is to print a message and abort the program, gsl: file.c:67: ERROR: invalid argument supplied by user Default GSL error handler invoked. Aborted The purpose of the `gsl_error' handler is to provide a function where a breakpoint can be set that will catch library errors when running under the debugger. It is not intended for use in production programs, which should handle any errors using the return codes.  File: gsl-ref.info, Node: Error Codes, Next: Error Handlers, Prev: Error Reporting, Up: Error Handling 3.2 Error Codes =============== The error code numbers returned by library functions are defined in the file `gsl_errno.h'. They all have the prefix `GSL_' and expand to non-zero constant integer values. Error codes above 1024 are reserved for applications, and are not used by the library. Many of the error codes use the same base name as the corresponding error code in the C library. Here are some of the most common error codes, -- Macro: int GSL_EDOM Domain error; used by mathematical functions when an argument value does not fall into the domain over which the function is defined (like EDOM in the C library) -- Macro: int GSL_ERANGE Range error; used by mathematical functions when the result value is not representable because of overflow or underflow (like ERANGE in the C library) -- Macro: int GSL_ENOMEM No memory available. The system cannot allocate more virtual memory because its capacity is full (like ENOMEM in the C library). This error is reported when a GSL routine encounters problems when trying to allocate memory with `malloc'. -- Macro: int GSL_EINVAL Invalid argument. This is used to indicate various kinds of problems with passing the wrong argument to a library function (like EINVAL in the C library). The error codes can be converted into an error message using the function `gsl_strerror'. -- Function: const char * gsl_strerror (const int GSL_ERRNO) This function returns a pointer to a string describing the error code GSL_ERRNO. For example, printf ("error: %s\n", gsl_strerror (status)); would print an error message like `error: output range error' for a status value of `GSL_ERANGE'.  File: gsl-ref.info, Node: Error Handlers, Next: Using GSL error reporting in your own functions, Prev: Error Codes, Up: Error Handling 3.3 Error Handlers ================== The default behavior of the GSL error handler is to print a short message and call `abort'. When this default is in use programs will stop with a core-dump whenever a library routine reports an error. This is intended as a fail-safe default for programs which do not check the return status of library routines (we don't encourage you to write programs this way). If you turn off the default error handler it is your responsibility to check the return values of routines and handle them yourself. You can also customize the error behavior by providing a new error handler. For example, an alternative error handler could log all errors to a file, ignore certain error conditions (such as underflows), or start the debugger and attach it to the current process when an error occurs. All GSL error handlers have the type `gsl_error_handler_t', which is defined in `gsl_errno.h', -- Data Type: gsl_error_handler_t This is the type of GSL error handler functions. An error handler will be passed four arguments which specify the reason for the error (a string), the name of the source file in which it occurred (also a string), the line number in that file (an integer) and the error number (an integer). The source file and line number are set at compile time using the `__FILE__' and `__LINE__' directives in the preprocessor. An error handler function returns type `void'. Error handler functions should be defined like this, void handler (const char * reason, const char * file, int line, int gsl_errno) To request the use of your own error handler you need to call the function `gsl_set_error_handler' which is also declared in `gsl_errno.h', -- Function: gsl_error_handler_t * gsl_set_error_handler (gsl_error_handler_t NEW_HANDLER) This function sets a new error handler, NEW_HANDLER, for the GSL library routines. The previous handler is returned (so that you can restore it later). Note that the pointer to a user defined error handler function is stored in a static variable, so there can be only one error handler per program. This function should be not be used in multi-threaded programs except to set up a program-wide error handler from a master thread. The following example shows how to set and restore a new error handler, /* save original handler, install new handler */ old_handler = gsl_set_error_handler (&my_handler); /* code uses new handler */ ..... /* restore original handler */ gsl_set_error_handler (old_handler); To use the default behavior (`abort' on error) set the error handler to `NULL', old_handler = gsl_set_error_handler (NULL); -- Function: gsl_error_handler_t * gsl_set_error_handler_off () This function turns off the error handler by defining an error handler which does nothing. This will cause the program to continue after any error, so the return values from any library routines must be checked. This is the recommended behavior for production programs. The previous handler is returned (so that you can restore it later). The error behavior can be changed for specific applications by recompiling the library with a customized definition of the `GSL_ERROR' macro in the file `gsl_errno.h'.  File: gsl-ref.info, Node: Using GSL error reporting in your own functions, Next: Error Reporting Examples, Prev: Error Handlers, Up: Error Handling 3.4 Using GSL error reporting in your own functions =================================================== If you are writing numerical functions in a program which also uses GSL code you may find it convenient to adopt the same error reporting conventions as in the library. To report an error you need to call the function `gsl_error' with a string describing the error and then return an appropriate error code from `gsl_errno.h', or a special value, such as `NaN'. For convenience the file `gsl_errno.h' defines two macros which carry out these steps: -- Macro: GSL_ERROR (REASON, GSL_ERRNO) This macro reports an error using the GSL conventions and returns a status value of `gsl_errno'. It expands to the following code fragment, gsl_error (reason, __FILE__, __LINE__, gsl_errno); return gsl_errno; The macro definition in `gsl_errno.h' actually wraps the code in a `do { ... } while (0)' block to prevent possible parsing problems. Here is an example of how the macro could be used to report that a routine did not achieve a requested tolerance. To report the error the routine needs to return the error code `GSL_ETOL'. if (residual > tolerance) { GSL_ERROR("residual exceeds tolerance", GSL_ETOL); } -- Macro: GSL_ERROR_VAL (REASON, GSL_ERRNO, VALUE) This macro is the same as `GSL_ERROR' but returns a user-defined value of VALUE instead of an error code. It can be used for mathematical functions that return a floating point value. The following example shows how to return a `NaN' at a mathematical singularity using the `GSL_ERROR_VAL' macro, if (x == 0) { GSL_ERROR_VAL("argument lies on singularity", GSL_ERANGE, GSL_NAN); }  File: gsl-ref.info, Node: Error Reporting Examples, Prev: Using GSL error reporting in your own functions, Up: Error Handling 3.5 Examples ============ Here is an example of some code which checks the return value of a function where an error might be reported, #include #include #include ... int status; size_t n = 37; gsl_set_error_handler_off(); status = gsl_fft_complex_radix2_forward (data, n); if (status) { if (status == GSL_EINVAL) { fprintf (stderr, "invalid argument, n=%d\n", n); } else { fprintf (stderr, "failed, gsl_errno=%d\n", status); } exit (-1); } ... The function `gsl_fft_complex_radix2' only accepts integer lengths which are a power of two. If the variable `n' is not a power of two then the call to the library function will return `GSL_EINVAL', indicating that the length argument is invalid. The function call to `gsl_set_error_handler_off' stops the default error handler from aborting the program. The `else' clause catches any other possible errors.  File: gsl-ref.info, Node: Mathematical Functions, Next: Complex Numbers, Prev: Error Handling, Up: Top 4 Mathematical Functions ************************ This chapter describes basic mathematical functions. Some of these functions are present in system libraries, but the alternative versions given here can be used as a substitute when the system functions are not available. The functions and macros described in this chapter are defined in the header file `gsl_math.h'. * Menu: * Mathematical Constants:: * Infinities and Not-a-number:: * Elementary Functions:: * Small integer powers:: * Testing the Sign of Numbers:: * Testing for Odd and Even Numbers:: * Maximum and Minimum functions:: * Approximate Comparison of Floating Point Numbers::  File: gsl-ref.info, Node: Mathematical Constants, Next: Infinities and Not-a-number, Up: Mathematical Functions 4.1 Mathematical Constants ========================== The library ensures that the standard BSD mathematical constants are defined. For reference, here is a list of the constants: `M_E' The base of exponentials, e `M_LOG2E' The base-2 logarithm of e, \log_2 (e) `M_LOG10E' The base-10 logarithm of e, \log_10 (e) `M_SQRT2' The square root of two, \sqrt 2 `M_SQRT1_2' The square root of one-half, \sqrt{1/2} `M_SQRT3' The square root of three, \sqrt 3 `M_PI' The constant pi, \pi `M_PI_2' Pi divided by two, \pi/2 `M_PI_4' Pi divided by four, \pi/4 `M_SQRTPI' The square root of pi, \sqrt\pi `M_2_SQRTPI' Two divided by the square root of pi, 2/\sqrt\pi `M_1_PI' The reciprocal of pi, 1/\pi `M_2_PI' Twice the reciprocal of pi, 2/\pi `M_LN10' The natural logarithm of ten, \ln(10) `M_LN2' The natural logarithm of two, \ln(2) `M_LNPI' The natural logarithm of pi, \ln(\pi) `M_EULER' Euler's constant, \gamma  File: gsl-ref.info, Node: Infinities and Not-a-number, Next: Elementary Functions, Prev: Mathematical Constants, Up: Mathematical Functions 4.2 Infinities and Not-a-number =============================== -- Macro: GSL_POSINF This macro contains the IEEE representation of positive infinity, +\infty. It is computed from the expression `+1.0/0.0'. -- Macro: GSL_NEGINF This macro contains the IEEE representation of negative infinity, -\infty. It is computed from the expression `-1.0/0.0'. -- Macro: GSL_NAN This macro contains the IEEE representation of the Not-a-Number symbol, `NaN'. It is computed from the ratio `0.0/0.0'. -- Function: int gsl_isnan (const double X) This function returns 1 if X is not-a-number. -- Function: int gsl_isinf (const double X) This function returns +1 if X is positive infinity, -1 if X is negative infinity and 0 otherwise. -- Function: int gsl_finite (const double X) This function returns 1 if X is a real number, and 0 if it is infinite or not-a-number.  File: gsl-ref.info, Node: Elementary Functions, Next: Small integer powers, Prev: Infinities and Not-a-number, Up: Mathematical Functions 4.3 Elementary Functions ======================== The following routines provide portable implementations of functions found in the BSD math library. When native versions are not available the functions described here can be used instead. The substitution can be made automatically if you use `autoconf' to compile your application (*note Portability functions::). -- Function: double gsl_log1p (const double X) This function computes the value of \log(1+x) in a way that is accurate for small X. It provides an alternative to the BSD math function `log1p(x)'. -- Function: double gsl_expm1 (const double X) This function computes the value of \exp(x)-1 in a way that is accurate for small X. It provides an alternative to the BSD math function `expm1(x)'. -- Function: double gsl_hypot (const double X, const double Y) This function computes the value of \sqrt{x^2 + y^2} in a way that avoids overflow. It provides an alternative to the BSD math function `hypot(x,y)'. -- Function: double gsl_acosh (const double X) This function computes the value of \arccosh(x). It provides an alternative to the standard math function `acosh(x)'. -- Function: double gsl_asinh (const double X) This function computes the value of \arcsinh(x). It provides an alternative to the standard math function `asinh(x)'. -- Function: double gsl_atanh (const double X) This function computes the value of \arctanh(x). It provides an alternative to the standard math function `atanh(x)'. -- Function: double gsl_ldexp (double X, int E) This function computes the value of x * 2^e. It provides an alternative to the standard math function `ldexp(x,e)'. -- Function: double gsl_frexp (double X, int * E) This function splits the number x into its normalized fraction f and exponent e, such that x = f * 2^e and 0.5 <= f < 1. The function returns f and stores the exponent in e. If x is zero, both f and e are set to zero. This function provides an alternative to the standard math function `frexp(x, e)'.  File: gsl-ref.info, Node: Small integer powers, Next: Testing the Sign of Numbers, Prev: Elementary Functions, Up: Mathematical Functions 4.4 Small integer powers ======================== A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides 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) This routine computes the power x^n for integer N. The power is computed efficiently--for example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications. A version of this function which also computes the numerical error in the result is available as `gsl_sf_pow_int_e'. -- Function: double gsl_pow_2 (const double X) -- Function: double gsl_pow_3 (const double X) -- Function: double gsl_pow_4 (const double X) -- Function: double gsl_pow_5 (const double X) -- Function: double gsl_pow_6 (const double X) -- Function: double gsl_pow_7 (const double X) -- Function: double gsl_pow_8 (const double X) -- Function: double gsl_pow_9 (const double X) These functions can be used to compute small integer powers x^2, x^3, etc. efficiently. The functions will be inlined when possible so that use of these functions should be as efficient as explicitly writing the corresponding product expression. #include double y = gsl_pow_4 (3.141) /* compute 3.141**4 */  File: gsl-ref.info, Node: Testing the Sign of Numbers, Next: Testing for Odd and Even Numbers, Prev: Small integer powers, Up: Mathematical Functions 4.5 Testing the Sign of Numbers =============================== -- Macro: GSL_SIGN (x) This macro returns the sign of X. It is defined as `((x) >= 0 ? 1 : -1)'. Note that with this definition the sign of zero is positive (regardless of its IEEE sign bit).  File: gsl-ref.info, Node: Testing for Odd and Even Numbers, Next: Maximum and Minimum functions, Prev: Testing the Sign of Numbers, Up: Mathematical Functions 4.6 Testing for Odd and Even Numbers ==================================== -- Macro: GSL_IS_ODD (n) This macro evaluates to 1 if N is odd and 0 if N is even. The argument N must be of integer type. -- Macro: GSL_IS_EVEN (n) This macro is the opposite of `GSL_IS_ODD(n)'. It evaluates to 1 if N is even and 0 if N is odd. The argument N must be of integer type.  File: gsl-ref.info, Node: Maximum and Minimum functions, Next: Approximate Comparison of Floating Point Numbers, Prev: Testing for Odd and Even Numbers, Up: Mathematical Functions 4.7 Maximum and Minimum functions ================================= -- Macro: GSL_MAX (a, b) This macro returns the maximum of A and B. It is defined as `((a) > (b) ? (a):(b))'. -- Macro: GSL_MIN (a, b) This macro returns the minimum of A and B. It is defined as `((a) < (b) ? (a):(b))'. -- Function: extern inline double GSL_MAX_DBL (double A, double B) This function returns the maximum of the double precision numbers A and B using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro `GSL_MAX' will be automatically substituted. -- Function: extern inline double GSL_MIN_DBL (double A, double B) This function returns the minimum of the double precision numbers A and B using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro `GSL_MIN' will be automatically substituted. -- Function: extern inline int GSL_MAX_INT (int A, int B) -- Function: extern inline int GSL_MIN_INT (int A, int B) These functions return the maximum or minimum of the integers A and B using an inline function. On platforms where inline functions are not available the macros `GSL_MAX' or `GSL_MIN' will be automatically substituted. -- Function: extern inline long double GSL_MAX_LDBL (long double A, long double B) -- Function: extern inline long double GSL_MIN_LDBL (long double A, long double B) These functions return the maximum or minimum of the long doubles A and B using an inline function. On platforms where inline functions are not available the macros `GSL_MAX' or `GSL_MIN' will be automatically substituted.  File: gsl-ref.info, Node: Approximate Comparison of Floating Point Numbers, Prev: Maximum and Minimum functions, Up: Mathematical Functions 4.8 Approximate Comparison of Floating Point Numbers ==================================================== It is sometimes useful to be able to compare two floating point numbers approximately, to allow for rounding and truncation errors. The following function implements the approximate floating-point comparison algorithm proposed by D.E. Knuth in Section 4.2.2 of `Seminumerical Algorithms' (3rd edition). -- Function: int gsl_fcmp (double X, double Y, double EPSILON) This function determines whether x and y are approximately equal to a relative accuracy EPSILON. The relative accuracy is measured using an interval of size 2 \delta, where \delta = 2^k \epsilon and k is the maximum base-2 exponent of x and y as computed by the function `frexp'. If x and y lie within this interval, they are considered approximately equal and the function returns 0. Otherwise if x < y, the function returns -1, or if x > y, the function returns +1. The implementation is based on the package `fcmp' by T.C. Belding.  File: gsl-ref.info, Node: Complex Numbers, Next: Polynomials, Prev: Mathematical Functions, Up: Top 5 Complex Numbers ***************** The functions described in this chapter provide support for complex numbers. The algorithms take care to avoid unnecessary intermediate underflows and overflows, allowing the functions to be evaluated over as much of the complex plane as possible. For multiple-valued functions the branch cuts have been chosen to follow the conventions of Abramowitz and Stegun in the `Handbook of Mathematical Functions'. The functions return principal values which are the same as those in GNU Calc, which in turn are the same as those in `Common Lisp, The Language (Second Edition)'(1) and the HP-28/48 series of calculators. The complex types are defined in the header file `gsl_complex.h', while the corresponding complex functions and arithmetic operations are defined in `gsl_complex_math.h'. * Menu: * Complex numbers:: * Properties of complex numbers:: * Complex arithmetic operators:: * Elementary Complex Functions:: * Complex Trigonometric Functions:: * Inverse Complex Trigonometric Functions:: * Complex Hyperbolic Functions:: * Inverse Complex Hyperbolic Functions:: * Complex Number References and Further Reading:: ---------- Footnotes ---------- (1) Note that the first edition uses different definitions.  File: gsl-ref.info, Node: Complex numbers, Next: Properties of complex numbers, Up: Complex Numbers 5.1 Complex numbers =================== Complex numbers are represented using the type `gsl_complex'. The internal representation of this type may vary across platforms and should not be accessed directly. The functions and macros described below allow complex numbers to be manipulated in a portable way. For reference, the default form of the `gsl_complex' type is given by the following struct, typedef struct { double dat[2]; } gsl_complex; The real and imaginary part are stored in contiguous elements of a two element array. This eliminates any padding between the real and imaginary parts, `dat[0]' and `dat[1]', allowing the struct to be mapped correctly onto packed complex arrays. -- Function: gsl_complex gsl_complex_rect (double X, double Y) This function uses the rectangular cartesian components (X,Y) to return the complex number z = x + i y. -- Function: gsl_complex gsl_complex_polar (double R, double THETA) This function returns the complex number z = r \exp(i \theta) = r (\cos(\theta) + i \sin(\theta)) from the polar representation (R,THETA). -- Macro: GSL_REAL (Z) -- Macro: GSL_IMAG (Z) These macros return the real and imaginary parts of the complex number Z. -- Macro: GSL_SET_COMPLEX (ZP, X, Y) This macro uses the cartesian components (X,Y) to set the real and imaginary parts of the complex number pointed to by ZP. For example, GSL_SET_COMPLEX(&z, 3, 4) sets Z to be 3 + 4i. -- Macro: GSL_SET_REAL (ZP,X) -- Macro: GSL_SET_IMAG (ZP,Y) These macros allow the real and imaginary parts of the complex number pointed to by ZP to be set independently.  File: gsl-ref.info, Node: Properties of complex numbers, Next: Complex arithmetic operators, Prev: Complex numbers, Up: Complex Numbers 5.2 Properties of complex numbers ================================= -- Function: double gsl_complex_arg (gsl_complex Z) This function returns the argument of the complex number Z, \arg(z), where -\pi < \arg(z) <= \pi. -- Function: double gsl_complex_abs (gsl_complex Z) This function returns the magnitude of the complex number Z, |z|. -- Function: double gsl_complex_abs2 (gsl_complex Z) This function returns the squared magnitude of the complex number Z, |z|^2. -- Function: double gsl_complex_logabs (gsl_complex Z) This function returns the natural logarithm of the magnitude of the complex number Z, \log|z|. It allows an accurate evaluation of \log|z| when |z| is close to one. The direct evaluation of `log(gsl_complex_abs(z))' would lead to a loss of precision in this case.  File: gsl-ref.info, Node: Complex arithmetic operators, Next: Elementary Complex Functions, Prev: Properties of complex numbers, Up: Complex Numbers 5.3 Complex arithmetic operators ================================ -- Function: gsl_complex gsl_complex_add (gsl_complex A, gsl_complex B) This function returns the sum of the complex numbers A and B, z=a+b. -- Function: gsl_complex gsl_complex_sub (gsl_complex A, gsl_complex B) This function returns the difference of the complex numbers A and B, z=a-b. -- Function: gsl_complex gsl_complex_mul (gsl_complex A, gsl_complex B) This function returns the product of the complex numbers A and B, z=ab. -- Function: gsl_complex gsl_complex_div (gsl_complex A, gsl_complex B) This function returns the quotient of the complex numbers A and B, z=a/b. -- Function: gsl_complex gsl_complex_add_real (gsl_complex A, double X) This function returns the sum of the complex number A and the real number X, z=a+x. -- Function: gsl_complex gsl_complex_sub_real (gsl_complex A, double X) This function returns the difference of the complex number A and the real number X, z=a-x. -- Function: gsl_complex gsl_complex_mul_real (gsl_complex A, double X) This function returns the product of the complex number A and the real number X, z=ax. -- Function: gsl_complex gsl_complex_div_real (gsl_complex A, double X) This function returns the quotient of the complex number A and the real number X, z=a/x. -- Function: gsl_complex gsl_complex_add_imag (gsl_complex A, double Y) This function returns the sum of the complex number A and the imaginary number iY, z=a+iy. -- Function: gsl_complex gsl_complex_sub_imag (gsl_complex A, double Y) This function returns the difference of the complex number A and the imaginary number iY, z=a-iy. -- Function: gsl_complex gsl_complex_mul_imag (gsl_complex A, double Y) This function returns the product of the complex number A and the imaginary number iY, z=a*(iy). -- Function: gsl_complex gsl_complex_div_imag (gsl_complex A, double Y) This function returns the quotient of the complex number A and the imaginary number iY, z=a/(iy). -- Function: gsl_complex gsl_complex_conjugate (gsl_complex Z) This function returns the complex conjugate of the complex number Z, z^* = x - i y. -- Function: gsl_complex gsl_complex_inverse (gsl_complex Z) This function returns the inverse, or reciprocal, of the complex number Z, 1/z = (x - i y)/(x^2 + y^2). -- Function: gsl_complex gsl_complex_negative (gsl_complex Z) This function returns the negative of the complex number Z, -z = (-x) + i(-y).  File: gsl-ref.info, Node: Elementary Complex Functions, Next: Complex Trigonometric Functions, Prev: Complex arithmetic operators, Up: Complex Numbers 5.4 Elementary Complex Functions ================================ -- Function: gsl_complex gsl_complex_sqrt (gsl_complex Z) This function returns the square root of the complex number Z, \sqrt z. The branch cut is the negative real axis. The result always lies in the right half of the complex plane. -- Function: gsl_complex gsl_complex_sqrt_real (double X) This function returns the complex square root of the real number X, where X may be negative. -- Function: gsl_complex gsl_complex_pow (gsl_complex Z, gsl_complex A) The function returns the complex number Z raised to the complex power A, z^a. This is computed as \exp(\log(z)*a) using complex logarithms and complex exponentials. -- Function: gsl_complex gsl_complex_pow_real (gsl_complex Z, double X) This function returns the complex number Z raised to the real power X, z^x. -- Function: gsl_complex gsl_complex_exp (gsl_complex Z) This function returns the complex exponential of the complex number Z, \exp(z). -- Function: gsl_complex gsl_complex_log (gsl_complex Z) This function returns the complex natural logarithm (base e) of the complex number Z, \log(z). The branch cut is the negative real axis. -- Function: gsl_complex gsl_complex_log10 (gsl_complex Z) This function returns the complex base-10 logarithm of the complex number Z, \log_10 (z). -- Function: gsl_complex gsl_complex_log_b (gsl_complex Z, gsl_complex B) This function returns the complex base-B logarithm of the complex number Z, \log_b(z). This quantity is computed as the ratio \log(z)/\log(b).  File: gsl-ref.info, Node: Complex Trigonometric Functions, Next: Inverse Complex Trigonometric Functions, Prev: Elementary Complex Functions, Up: Complex Numbers 5.5 Complex Trigonometric Functions =================================== -- Function: gsl_complex gsl_complex_sin (gsl_complex Z) This function returns the complex sine of the complex number Z, \sin(z) = (\exp(iz) - \exp(-iz))/(2i). -- Function: gsl_complex gsl_complex_cos (gsl_complex Z) This function returns the complex cosine of the complex number Z, \cos(z) = (\exp(iz) + \exp(-iz))/2. -- Function: gsl_complex gsl_complex_tan (gsl_complex Z) This function returns the complex tangent of the complex number Z, \tan(z) = \sin(z)/\cos(z). -- Function: gsl_complex gsl_complex_sec (gsl_complex Z) This function returns the complex secant of the complex number Z, \sec(z) = 1/\cos(z). -- Function: gsl_complex gsl_complex_csc (gsl_complex Z) This function returns the complex cosecant of the complex number Z, \csc(z) = 1/\sin(z). -- Function: gsl_complex gsl_complex_cot (gsl_complex Z) This function returns the complex cotangent of the complex number Z, \cot(z) = 1/\tan(z).  File: gsl-ref.info, Node: Inverse Complex Trigonometric Functions, Next: Complex Hyperbolic Functions, Prev: Complex Trigonometric Functions, Up: Complex Numbers 5.6 Inverse Complex Trigonometric Functions =========================================== -- Function: gsl_complex gsl_complex_arcsin (gsl_complex Z) This function returns the complex arcsine of the complex number Z, \arcsin(z). The branch cuts are on the real axis, less than -1 and greater than 1. -- Function: gsl_complex gsl_complex_arcsin_real (double Z) This function returns the complex arcsine of the real number Z, \arcsin(z). For z between -1 and 1, the function returns a real value in the range [-\pi/2,\pi/2]. For z less than -1 the result has a real part of -\pi/2 and a positive imaginary part. For z greater than 1 the result has a real part of \pi/2 and a negative imaginary part. -- Function: gsl_complex gsl_complex_arccos (gsl_complex Z) This function returns the complex arccosine of the complex number Z, \arccos(z). The branch cuts are on the real axis, less than -1 and greater than 1. -- Function: gsl_complex gsl_complex_arccos_real (double Z) This function returns the complex arccosine of the real number Z, \arccos(z). For z between -1 and 1, the function returns a real value in the range [0,\pi]. For z less than -1 the result has a real part of \pi and a negative imaginary part. For z greater than 1 the result is purely imaginary and positive. -- Function: gsl_complex gsl_complex_arctan (gsl_complex Z) This function returns the complex arctangent of the complex number Z, \arctan(z). The branch cuts are on the imaginary axis, below -i and above i. -- Function: gsl_complex gsl_complex_arcsec (gsl_complex Z) This function returns the complex arcsecant of the complex number Z, \arcsec(z) = \arccos(1/z). -- Function: gsl_complex gsl_complex_arcsec_real (double Z) This function returns the complex arcsecant of the real number Z, \arcsec(z) = \arccos(1/z). -- Function: gsl_complex gsl_complex_arccsc (gsl_complex Z) This function returns the complex arccosecant of the complex number Z, \arccsc(z) = \arcsin(1/z). -- Function: gsl_complex gsl_complex_arccsc_real (double Z) This function returns the complex arccosecant of the real number Z, \arccsc(z) = \arcsin(1/z). -- Function: gsl_complex gsl_complex_arccot (gsl_complex Z) This function returns the complex arccotangent of the complex number Z, \arccot(z) = \arctan(1/z).  File: gsl-ref.info, Node: Complex Hyperbolic Functions, Next: Inverse Complex Hyperbolic Functions, Prev: Inverse Complex Trigonometric Functions, Up: Complex Numbers 5.7 Complex Hyperbolic Functions ================================ -- Function: gsl_complex gsl_complex_sinh (gsl_complex Z) This function returns the complex hyperbolic sine of the complex number Z, \sinh(z) = (\exp(z) - \exp(-z))/2. -- Function: gsl_complex gsl_complex_cosh (gsl_complex Z) This function returns the complex hyperbolic cosine of the complex number Z, \cosh(z) = (\exp(z) + \exp(-z))/2. -- Function: gsl_complex gsl_complex_tanh (gsl_complex Z) This function returns the complex hyperbolic tangent of the complex number Z, \tanh(z) = \sinh(z)/\cosh(z). -- Function: gsl_complex gsl_complex_sech (gsl_complex Z) This function returns the complex hyperbolic secant of the complex number Z, \sech(z) = 1/\cosh(z). -- Function: gsl_complex gsl_complex_csch (gsl_complex Z) This function returns the complex hyperbolic cosecant of the complex number Z, \csch(z) = 1/\sinh(z). -- Function: gsl_complex gsl_complex_coth (gsl_complex Z) This function returns the complex hyperbolic cotangent of the complex number Z, \coth(z) = 1/\tanh(z).  File: gsl-ref.info, Node: Inverse Complex Hyperbolic Functions, Next: Complex Number References and Further Reading, Prev: Complex Hyperbolic Functions, Up: Complex Numbers 5.8 Inverse Complex Hyperbolic Functions ======================================== -- Function: gsl_complex gsl_complex_arcsinh (gsl_complex Z) This function returns the complex hyperbolic arcsine of the complex number Z, \arcsinh(z). The branch cuts are on the imaginary axis, below -i and above i. -- Function: gsl_complex gsl_complex_arccosh (gsl_complex Z) This function returns the complex hyperbolic arccosine of the complex number Z, \arccosh(z). The branch cut is on the real axis, less than 1. Note that in this case we use the negative square root in formula 4.6.21 of Abramowitz & Stegun giving \arccosh(z)=\log(z-\sqrt{z^2-1}). -- Function: gsl_complex gsl_complex_arccosh_real (double Z) This function returns the complex hyperbolic arccosine of the real number Z, \arccosh(z). -- Function: gsl_complex gsl_complex_arctanh (gsl_complex Z) This function returns the complex hyperbolic arctangent of the complex number Z, \arctanh(z). The branch cuts are on the real axis, less than -1 and greater than 1. -- Function: gsl_complex gsl_complex_arctanh_real (double Z) This function returns the complex hyperbolic arctangent of the real number Z, \arctanh(z). -- Function: gsl_complex gsl_complex_arcsech (gsl_complex Z) This function returns the complex hyperbolic arcsecant of the complex number Z, \arcsech(z) = \arccosh(1/z). -- Function: gsl_complex gsl_complex_arccsch (gsl_complex Z) This function returns the complex hyperbolic arccosecant of the complex number Z, \arccsch(z) = \arcsin(1/z). -- Function: gsl_complex gsl_complex_arccoth (gsl_complex Z) This function returns the complex hyperbolic arccotangent of the complex number Z, \arccoth(z) = \arctanh(1/z).  File: gsl-ref.info, Node: Complex Number References and Further Reading, Prev: Inverse Complex Hyperbolic Functions, Up: Complex Numbers 5.9 References and Further Reading ================================== The implementations of the elementary and trigonometric functions are based on the following papers, T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang, "Implementing Complex Elementary Functions Using Exception Handling", `ACM Transactions on Mathematical Software', Volume 20 (1994), pp 215-244, Corrigenda, p553 T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang, "Implementing the complex arcsin and arccosine functions using exception handling", `ACM Transactions on Mathematical Software', Volume 23 (1997) pp 299-335 The general formulas and details of branch cuts can be found in the following books, Abramowitz and Stegun, `Handbook of Mathematical Functions', "Circular Functions in Terms of Real and Imaginary Parts", Formulas 4.3.55-58, "Inverse Circular Functions in Terms of Real and Imaginary Parts", Formulas 4.4.37-39, "Hyperbolic Functions in Terms of Real and Imaginary Parts", Formulas 4.5.49-52, "Inverse Hyperbolic Functions--relation to Inverse Circular Functions", Formulas 4.6.14-19. Dave Gillespie, `Calc Manual', Free Software Foundation, ISBN 1-882114-18-3  File: gsl-ref.info, Node: Polynomials, Next: Special Functions, Prev: Complex Numbers, Up: Top 6 Polynomials ************* This chapter describes functions for evaluating and solving polynomials. There are routines for finding real and complex roots of quadratic and cubic equations using analytic methods. An iterative polynomial solver is also available for finding the roots of general polynomials with real coefficients (of any order). The functions are declared in the header file `gsl_poly.h'. * Menu: * Polynomial Evaluation:: * Divided Difference Representation of Polynomials:: * Quadratic Equations:: * Cubic Equations:: * General Polynomial Equations:: * Roots of Polynomials Examples:: * Roots of Polynomials References and Further Reading::  File: gsl-ref.info, Node: Polynomial Evaluation, Next: Divided Difference Representation of Polynomials, Up: Polynomials 6.1 Polynomial Evaluation ========================= -- Function: double gsl_poly_eval (const double C[], const int LEN, const double X) This function evaluates the polynomial c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1} using Horner's method for stability. The function is inlined when possible.  File: gsl-ref.info, Node: Divided Difference Representation of Polynomials, Next: Quadratic Equations, Prev: Polynomial Evaluation, Up: Polynomials 6.2 Divided Difference Representation of Polynomials ==================================================== The functions described here manipulate polynomials stored in Newton's divided-difference representation. The use of divided-differences is described in Abramowitz & Stegun sections 25.1.4 and 25.2.26. -- Function: int gsl_poly_dd_init (double DD[], const double XA[], const double YA[], size_t SIZE) This function computes a divided-difference representation of the interpolating polynomial for the points (XA, YA) stored in the arrays XA and YA of length SIZE. On output the divided-differences of (XA,YA) are stored in the array DD, also of length SIZE. -- Function: double gsl_poly_dd_eval (const double DD[], const double XA[], const size_t SIZE, const double X) This function evaluates the polynomial stored in divided-difference form in the arrays DD and XA of length SIZE at the point X. -- Function: int gsl_poly_dd_taylor (double C[], double XP, const double DD[], const double XA[], size_t SIZE, double W[]) This function converts the divided-difference representation of a polynomial to a Taylor expansion. The divided-difference representation is supplied in the arrays DD and XA of length SIZE. On output the Taylor coefficients of the polynomial expanded about the point XP are stored in the array C also of length SIZE. A workspace of length SIZE must be provided in the array W.  File: gsl-ref.info, Node: Quadratic Equations, Next: Cubic Equations, Prev: Divided Difference Representation of Polynomials, Up: Polynomials 6.3 Quadratic Equations ======================= -- Function: int gsl_poly_solve_quadratic (double A, double B, double C, double * X0, double * X1) This function finds the real roots of the quadratic equation, a x^2 + b x + c = 0 The number of real roots (either zero, one or two) is returned, and their locations are stored in X0 and X1. If no real roots are found then X0 and X1 are not modified. If one real root is found (i.e. if a=0) then it is stored in X0. When two real roots are found they are stored in X0 and X1 in ascending order. The case of coincident roots is not considered special. For example (x-1)^2=0 will have two roots, which happen to have exactly equal values. The number of roots found depends on the sign of the discriminant b^2 - 4 a c. This will be subject to rounding and cancellation errors when computed in double precision, and will also be subject to errors if the coefficients of the polynomial are inexact. These errors may cause a discrete change in the number of roots. However, for polynomials with small integer coefficients the discriminant can always be computed exactly. -- Function: int gsl_poly_complex_solve_quadratic (double A, double B, double C, gsl_complex * Z0, gsl_complex * Z1) This function finds the complex roots of the quadratic equation, a z^2 + b z + c = 0 The number of complex roots is returned (either one or two) and the locations of the roots are stored in Z0 and Z1. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components. If only one real root is found (i.e. if a=0) then it is stored in Z0.  File: gsl-ref.info, Node: Cubic Equations, Next: General Polynomial Equations, Prev: Quadratic Equations, Up: Polynomials 6.4 Cubic Equations =================== -- Function: int gsl_poly_solve_cubic (double A, double B, double C, double * X0, double * X1, double * X2) This function finds the real roots of the cubic equation, x^3 + a x^2 + b x + c = 0 with a leading coefficient of unity. The number of real roots (either one or three) is returned, and their locations are stored in X0, X1 and X2. If one real root is found then only X0 is modified. When three real roots are found they are stored in X0, X1 and X2 in ascending order. The case of coincident roots is not considered special. For example, the equation (x-1)^3=0 will have three roots with exactly equal values. -- Function: int gsl_poly_complex_solve_cubic (double A, double B, double C, gsl_complex * Z0, gsl_complex * Z1, gsl_complex * Z2) This function finds the complex roots of the cubic equation, z^3 + a z^2 + b z + c = 0 The number of complex roots is returned (always three) and the locations of the roots are stored in Z0, Z1 and Z2. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components.  File: gsl-ref.info, Node: General Polynomial Equations, Next: Roots of Polynomials Examples, Prev: Cubic Equations, Up: Polynomials 6.5 General Polynomial Equations ================================ The roots of polynomial equations cannot be found analytically beyond the special cases of the quadratic, cubic and quartic equation. The algorithm described in this section uses an iterative method to find the approximate locations of roots of higher order polynomials. -- Function: gsl_poly_complex_workspace * gsl_poly_complex_workspace_alloc (size_t N) This function allocates space for a `gsl_poly_complex_workspace' struct and a workspace suitable for solving a polynomial with N coefficients using the routine `gsl_poly_complex_solve'. The function returns a pointer to the newly allocated `gsl_poly_complex_workspace' if no errors were detected, and a null pointer in the case of error. -- Function: void gsl_poly_complex_workspace_free (gsl_poly_complex_workspace * W) This function frees all the memory associated with the workspace W. -- Function: int gsl_poly_complex_solve (const double * A, size_t N, gsl_poly_complex_workspace * W, gsl_complex_packed_ptr Z) This function computes the roots of the general polynomial P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using balanced-QR reduction of the companion matrix. The parameter N specifies the length of the coefficient array. The coefficient of the highest order term must be non-zero. The function requires a workspace W of the appropriate size. The n-1 roots are returned in the packed complex array Z of length 2(n-1), alternating real and imaginary parts. The function returns `GSL_SUCCESS' if all the roots are found. If the QR reduction does not converge, the error handler is invoked with an error code of `GSL_EFAILED'. Note that due to finite precision, roots of higher multiplicity are returned as a cluster of simple roots with reduced accuracy. The solution of polynomials with higher-order roots requires specialized algorithms that take the multiplicity structure into account (see e.g. Z. Zeng, Algorithm 835, ACM Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp 218-236).  File: gsl-ref.info, Node: Roots of Polynomials Examples, Next: Roots of Polynomials References and Further Reading, Prev: General Polynomial Equations, Up: Polynomials 6.6 Examples ============ To demonstrate the use of the general polynomial solver we will take the polynomial P(x) = x^5 - 1 which has the following roots, 1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5} The following program will find these roots. #include #include int main (void) { int i; /* coefficients of P(x) = -1 + x^5 */ double a[6] = { -1, 0, 0, 0, 0, 1 }; double z[10]; gsl_poly_complex_workspace * w = gsl_poly_complex_workspace_alloc (6); gsl_poly_complex_solve (a, 6, w, z); gsl_poly_complex_workspace_free (w); for (i = 0; i < 5; i++) { printf ("z%d = %+.18f %+.18f\n", i, z[2*i], z[2*i+1]); } return 0; } The output of the program is, $ ./a.out z0 = -0.809016994374947451 +0.587785252292473137 z1 = -0.809016994374947451 -0.587785252292473137 z2 = +0.309016994374947451 +0.951056516295153642 z3 = +0.309016994374947451 -0.951056516295153642 z4 = +1.000000000000000000 +0.000000000000000000 which agrees with the analytic result, z_n = \exp(2 \pi n i/5).  File: gsl-ref.info, Node: Roots of Polynomials References and Further Reading, Prev: Roots of Polynomials Examples, Up: Polynomials 6.7 References and Further Reading ================================== The balanced-QR method and its error analysis are described in the following papers, R.S. Martin, G. Peters and J.H. Wilkinson, "The QR Algorithm for Real Hessenberg Matrices", `Numerische Mathematik', 14 (1970), 219-231. B.N. Parlett and C. Reinsch, "Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors", `Numerische Mathematik', 13 (1969), 293-304. A. Edelman and H. Murakami, "Polynomial roots from companion matrix eigenvalues", `Mathematics of Computation', Vol. 64, No. 210 (1995), 763-776. The formulas for divided differences are given in Abramowitz and Stegun, Abramowitz and Stegun, `Handbook of Mathematical Functions', Sections 25.1.4 and 25.2.26.  File: gsl-ref.info, Node: Special Functions, Next: Vectors and Matrices, Prev: Polynomials, Up: Top 7 Special Functions ******************* This chapter describes the GSL special function library. The library includes routines for calculating the values of Airy functions, Bessel functions, Clausen functions, Coulomb wave functions, Coupling coefficients, the Dawson function, Debye functions, Dilogarithms, Elliptic integrals, Jacobi elliptic functions, Error functions, Exponential integrals, Fermi-Dirac functions, Gamma functions, Gegenbauer functions, Hypergeometric functions, Laguerre functions, Legendre functions and Spherical Harmonics, the Psi (Digamma) Function, Synchrotron functions, Transport functions, Trigonometric functions and Zeta functions. Each routine also computes an estimate of the numerical error in the calculated value of the function. The functions in this chapter are declared in individual header files, such as `gsl_sf_airy.h', `gsl_sf_bessel.h', etc. The complete set of header files can be included using the file `gsl_sf.h'. * Menu: * Special Function Usage:: * The gsl_sf_result struct:: * Special Function Modes:: * Airy Functions and Derivatives:: * Bessel Functions:: * Clausen Functions:: * Coulomb Functions:: * Coupling Coefficients:: * Dawson Function:: * Debye Functions:: * Dilogarithm:: * Elementary Operations:: * Elliptic Integrals:: * Elliptic Functions (Jacobi):: * Error Functions:: * Exponential Functions:: * Exponential Integrals:: * Fermi-Dirac Function:: * Gamma and Beta Functions:: * Gegenbauer Functions:: * Hypergeometric Functions:: * Laguerre Functions:: * Lambert W Functions:: * Legendre Functions and Spherical Harmonics:: * Logarithm and Related Functions:: * Mathieu Functions:: * Power Function:: * Psi (Digamma) Function:: * Synchrotron Functions:: * Transport Functions:: * Trigonometric Functions:: * Zeta Functions:: * Special Functions Examples:: * Special Functions References and Further Reading::  File: gsl-ref.info, Node: Special Function Usage, Next: The gsl_sf_result struct, Up: Special Functions 7.1 Usage ========= The special functions are available in two calling conventions, a "natural form" which returns the numerical value of the function and an "error-handling form" which returns an error code. The two types of function provide alternative ways of accessing the same underlying code. The "natural form" returns only the value of the function and can be used directly in mathematical expressions. For example, the following function call will compute the value of the Bessel function J_0(x), double y = gsl_sf_bessel_J0 (x); There is no way to access an error code or to estimate the error using this method. To allow access to this information the alternative error-handling form stores the value and error in a modifiable argument, gsl_sf_result result; int status = gsl_sf_bessel_J0_e (x, &result); The error-handling functions have the suffix `_e'. The returned status value indicates error conditions such as overflow, underflow or loss of precision. If there are no errors the error-handling functions return `GSL_SUCCESS'.  File: gsl-ref.info, Node: The gsl_sf_result struct, Next: Special Function Modes, Prev: Special Function Usage, Up: Special Functions 7.2 The gsl_sf_result struct ============================ The error handling form of the special functions always calculate an error estimate along with the value of the result. Therefore, structures are provided for amalgamating a value and error estimate. These structures are declared in the header file `gsl_sf_result.h'. The `gsl_sf_result' struct contains value and error fields. typedef struct { double val; double err; } gsl_sf_result; The field VAL contains the value and the field ERR contains an estimate of the absolute error in the value. In some cases, an overflow or underflow can be detected and handled by a function. In this case, it may be possible to return a scaling exponent as well as an error/value pair in order to save the result from exceeding the dynamic range of the built-in types. The `gsl_sf_result_e10' struct contains value and error fields as well as an exponent field such that the actual result is obtained as `result * 10^(e10)'. typedef struct { double val; double err; int e10; } gsl_sf_result_e10;  File: gsl-ref.info, Node: Special Function Modes, Next: Airy Functions and Derivatives, Prev: The gsl_sf_result struct, Up: Special Functions 7.3 Modes ========= The goal of the library is to achieve double precision accuracy wherever possible. However the cost of evaluating some special functions to double precision can be significant, particularly where very high order terms are required. In these cases a `mode' argument allows the accuracy of the function to be reduced in order to improve performance. The following precision levels are available for the mode argument, `GSL_PREC_DOUBLE' Double-precision, a relative accuracy of approximately 2 * 10^-16. `GSL_PREC_SINGLE' Single-precision, a relative accuracy of approximately 10^-7. `GSL_PREC_APPROX' Approximate values, a relative accuracy of approximately 5 * 10^-4. The approximate mode provides the fastest evaluation at the lowest accuracy.  File: gsl-ref.info, Node: Airy Functions and Derivatives, Next: Bessel Functions, Prev: Special Function Modes, Up: Special Functions 7.4 Airy Functions and Derivatives ================================== The Airy functions Ai(x) and Bi(x) are defined by the integral representations, Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt For further information see Abramowitz & Stegun, Section 10.4. The Airy functions are defined in the header file `gsl_sf_airy.h'. * Menu: * Airy Functions:: * Derivatives of Airy Functions:: * Zeros of Airy Functions:: * Zeros of Derivatives of Airy Functions::  File: gsl-ref.info, Node: Airy Functions, Next: Derivatives of Airy Functions, Up: Airy Functions and Derivatives 7.4.1 Airy Functions -------------------- -- Function: double gsl_sf_airy_Ai (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Ai_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the Airy function Ai(x) with an accuracy specified by MODE. -- Function: double gsl_sf_airy_Bi (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Bi_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the Airy function Bi(x) with an accuracy specified by MODE. -- Function: double gsl_sf_airy_Ai_scaled (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Ai_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute a scaled version of the Airy function S_A(x) Ai(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)), and is 1 for x<0. -- Function: double gsl_sf_airy_Bi_scaled (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Bi_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute a scaled version of the Airy function S_B(x) Bi(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0.  File: gsl-ref.info, Node: Derivatives of Airy Functions, Next: Zeros of Airy Functions, Prev: Airy Functions, Up: Airy Functions and Derivatives 7.4.2 Derivatives of Airy Functions ----------------------------------- -- Function: double gsl_sf_airy_Ai_deriv (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Ai_deriv_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the Airy function derivative Ai'(x) with an accuracy specified by MODE. -- Function: double gsl_sf_airy_Bi_deriv (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Bi_deriv_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the Airy function derivative Bi'(x) with an accuracy specified by MODE. -- Function: double gsl_sf_airy_Ai_deriv_scaled (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Ai_deriv_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the scaled Airy function derivative S_A(x) Ai'(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)), and is 1 for x<0. -- Function: double gsl_sf_airy_Bi_deriv_scaled (double X, gsl_mode_t MODE) -- Function: int gsl_sf_airy_Bi_deriv_scaled_e (double X, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the scaled Airy function derivative S_B(x) Bi'(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0.  File: gsl-ref.info, Node: Zeros of Airy Functions, Next: Zeros of Derivatives of Airy Functions, Prev: Derivatives of Airy Functions, Up: Airy Functions and Derivatives 7.4.3 Zeros of Airy Functions ----------------------------- -- Function: double gsl_sf_airy_zero_Ai (unsigned int S) -- Function: int gsl_sf_airy_zero_Ai_e (unsigned int S, gsl_sf_result * RESULT) These routines compute the location of the S-th zero of the Airy function Ai(x). -- Function: double gsl_sf_airy_zero_Bi (unsigned int S) -- Function: int gsl_sf_airy_zero_Bi_e (unsigned int S, gsl_sf_result * RESULT) These routines compute the location of the S-th zero of the Airy function Bi(x).  File: gsl-ref.info, Node: Zeros of Derivatives of Airy Functions, Prev: Zeros of Airy Functions, Up: Airy Functions and Derivatives 7.4.4 Zeros of Derivatives of Airy Functions -------------------------------------------- -- Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int S) -- Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int S, gsl_sf_result * RESULT) These routines compute the location of the S-th zero of the Airy function derivative Ai'(x). -- Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int S) -- Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int S, gsl_sf_result * RESULT) These routines compute the location of the S-th zero of the Airy function derivative Bi'(x).  File: gsl-ref.info, Node: Bessel Functions, Next: Clausen Functions, Prev: Airy Functions and Derivatives, Up: Special Functions 7.5 Bessel Functions ==================== The routines described in this section compute the Cylindrical Bessel functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x), K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified Spherical Bessel functions i_l(x), k_l(x). For more information see Abramowitz & Stegun, Chapters 9 and 10. The Bessel functions are defined in the header file `gsl_sf_bessel.h'. * Menu: * Regular Cylindrical Bessel Functions:: * Irregular Cylindrical Bessel Functions:: * Regular Modified Cylindrical Bessel Functions:: * Irregular Modified Cylindrical Bessel Functions:: * Regular Spherical Bessel Functions:: * Irregular Spherical Bessel Functions:: * Regular Modified Spherical Bessel Functions:: * Irregular Modified Spherical Bessel Functions:: * Regular Bessel Function - Fractional Order:: * Irregular Bessel Functions - Fractional Order:: * Regular Modified Bessel Functions - Fractional Order:: * Irregular Modified Bessel Functions - Fractional Order:: * Zeros of Regular Bessel Functions::  File: gsl-ref.info, Node: Regular Cylindrical Bessel Functions, Next: Irregular Cylindrical Bessel Functions, Up: Bessel Functions 7.5.1 Regular Cylindrical Bessel Functions ------------------------------------------ -- Function: double gsl_sf_bessel_J0 (double X) -- Function: int gsl_sf_bessel_J0_e (double X, gsl_sf_result * RESULT) These routines compute the regular cylindrical Bessel function of zeroth order, J_0(x). -- Function: double gsl_sf_bessel_J1 (double X) -- Function: int gsl_sf_bessel_J1_e (double X, gsl_sf_result * RESULT) These routines compute the regular cylindrical Bessel function of first order, J_1(x). -- Function: double gsl_sf_bessel_Jn (int N, double X) -- Function: int gsl_sf_bessel_Jn_e (int N, double X, gsl_sf_result * RESULT) These routines compute the regular cylindrical Bessel function of order N, J_n(x). -- Function: int gsl_sf_bessel_Jn_array (int NMIN, int NMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the regular cylindrical Bessel functions J_n(x) for n from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.  File: gsl-ref.info, Node: Irregular Cylindrical Bessel Functions, Next: Regular Modified Cylindrical Bessel Functions, Prev: Regular Cylindrical Bessel Functions, Up: Bessel Functions 7.5.2 Irregular Cylindrical Bessel Functions -------------------------------------------- -- Function: double gsl_sf_bessel_Y0 (double X) -- Function: int gsl_sf_bessel_Y0_e (double X, gsl_sf_result * RESULT) These routines compute the irregular cylindrical Bessel function of zeroth order, Y_0(x), for x>0. -- Function: double gsl_sf_bessel_Y1 (double X) -- Function: int gsl_sf_bessel_Y1_e (double X, gsl_sf_result * RESULT) These routines compute the irregular cylindrical Bessel function of first order, Y_1(x), for x>0. -- Function: double gsl_sf_bessel_Yn (int N,double X) -- Function: int gsl_sf_bessel_Yn_e (int N,double X, gsl_sf_result * RESULT) These routines compute the irregular cylindrical Bessel function of order N, Y_n(x), for x>0. -- Function: int gsl_sf_bessel_Yn_array (int NMIN, int NMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the irregular cylindrical Bessel functions Y_n(x) for n from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY. The domain of the function is x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.  File: gsl-ref.info, Node: Regular Modified Cylindrical Bessel Functions, Next: Irregular Modified Cylindrical Bessel Functions, Prev: Irregular Cylindrical Bessel Functions, Up: Bessel Functions 7.5.3 Regular Modified Cylindrical Bessel Functions --------------------------------------------------- -- Function: double gsl_sf_bessel_I0 (double X) -- Function: int gsl_sf_bessel_I0_e (double X, gsl_sf_result * RESULT) These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x). -- Function: double gsl_sf_bessel_I1 (double X) -- Function: int gsl_sf_bessel_I1_e (double X, gsl_sf_result * RESULT) These routines compute the regular modified cylindrical Bessel function of first order, I_1(x). -- Function: double gsl_sf_bessel_In (int N, double X) -- Function: int gsl_sf_bessel_In_e (int N, double X, gsl_sf_result * RESULT) These routines compute the regular modified cylindrical Bessel function of order N, I_n(x). -- Function: int gsl_sf_bessel_In_array (int NMIN, int NMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the regular modified cylindrical Bessel functions I_n(x) for n from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY. The start of the range NMIN must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. -- Function: double gsl_sf_bessel_I0_scaled (double X) -- Function: int gsl_sf_bessel_I0_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x). -- Function: double gsl_sf_bessel_I1_scaled (double X) -- Function: int gsl_sf_bessel_I1_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x). -- Function: double gsl_sf_bessel_In_scaled (int N, double X) -- Function: int gsl_sf_bessel_In_scaled_e (int N, double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified cylindrical Bessel function of order N, \exp(-|x|) I_n(x) -- Function: int gsl_sf_bessel_In_scaled_array (int NMIN, int NMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the scaled regular cylindrical Bessel functions \exp(-|x|) I_n(x) for n from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY. The start of the range NMIN must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.  File: gsl-ref.info, Node: Irregular Modified Cylindrical Bessel Functions, Next: Regular Spherical Bessel Functions, Prev: Regular Modified Cylindrical Bessel Functions, Up: Bessel Functions 7.5.4 Irregular Modified Cylindrical Bessel Functions ----------------------------------------------------- -- Function: double gsl_sf_bessel_K0 (double X) -- Function: int gsl_sf_bessel_K0_e (double X, gsl_sf_result * RESULT) These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0. -- Function: double gsl_sf_bessel_K1 (double X) -- Function: int gsl_sf_bessel_K1_e (double X, gsl_sf_result * RESULT) These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0. -- Function: double gsl_sf_bessel_Kn (int N, double X) -- Function: int gsl_sf_bessel_Kn_e (int N, double X, gsl_sf_result * RESULT) These routines compute the irregular modified cylindrical Bessel function of order N, K_n(x), for x > 0. -- Function: int gsl_sf_bessel_Kn_array (int NMIN, int NMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the irregular modified cylindrical Bessel functions K_n(x) for n from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY. The start of the range NMIN must be positive or zero. The domain of the function is x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. -- Function: double gsl_sf_bessel_K0_scaled (double X) -- Function: int gsl_sf_bessel_K0_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0. -- Function: double gsl_sf_bessel_K1_scaled (double X) -- Function: int gsl_sf_bessel_K1_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified cylindrical Bessel function of first order \exp(x) K_1(x) for x>0. -- Function: double gsl_sf_bessel_Kn_scaled (int N, double X) -- Function: int gsl_sf_bessel_Kn_scaled_e (int N, double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified cylindrical Bessel function of order N, \exp(x) K_n(x), for x>0. -- Function: int gsl_sf_bessel_Kn_scaled_array (int NMIN, int NMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the scaled irregular cylindrical Bessel functions \exp(x) K_n(x) for n from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY. The start of the range NMIN must be positive or zero. The domain of the function is x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.  File: gsl-ref.info, Node: Regular Spherical Bessel Functions, Next: Irregular Spherical Bessel Functions, Prev: Irregular Modified Cylindrical Bessel Functions, Up: Bessel Functions 7.5.5 Regular Spherical Bessel Functions ---------------------------------------- -- Function: double gsl_sf_bessel_j0 (double X) -- Function: int gsl_sf_bessel_j0_e (double X, gsl_sf_result * RESULT) These routines compute the regular spherical Bessel function of zeroth order, j_0(x) = \sin(x)/x. -- Function: double gsl_sf_bessel_j1 (double X) -- Function: int gsl_sf_bessel_j1_e (double X, gsl_sf_result * RESULT) These routines compute the regular spherical Bessel function of first order, j_1(x) = (\sin(x)/x - \cos(x))/x. -- Function: double gsl_sf_bessel_j2 (double X) -- Function: int gsl_sf_bessel_j2_e (double X, gsl_sf_result * RESULT) These routines compute the regular spherical Bessel function of second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x. -- Function: double gsl_sf_bessel_jl (int L, double X) -- Function: int gsl_sf_bessel_jl_e (int L, double X, gsl_sf_result * RESULT) These routines compute the regular spherical Bessel function of order L, j_l(x), for l >= 0 and x >= 0. -- Function: int gsl_sf_bessel_jl_array (int LMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the regular spherical Bessel functions j_l(x) for l from 0 to LMAX inclusive for lmax >= 0 and x >= 0, storing the results in the array RESULT_ARRAY. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. -- Function: int gsl_sf_bessel_jl_steed_array (int LMAX, double X, double * JL_X_ARRAY) This routine uses Steed's method to compute the values of the regular spherical Bessel functions j_l(x) for l from 0 to LMAX inclusive for lmax >= 0 and x >= 0, storing the results in the array RESULT_ARRAY. The Steed/Barnett algorithm is described in `Comp. Phys. Comm.' 21, 297 (1981). Steed's method is more stable than the recurrence used in the other functions but is also slower.  File: gsl-ref.info, Node: Irregular Spherical Bessel Functions, Next: Regular Modified Spherical Bessel Functions, Prev: Regular Spherical Bessel Functions, Up: Bessel Functions 7.5.6 Irregular Spherical Bessel Functions ------------------------------------------ -- Function: double gsl_sf_bessel_y0 (double X) -- Function: int gsl_sf_bessel_y0_e (double X, gsl_sf_result * RESULT) These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x. -- Function: double gsl_sf_bessel_y1 (double X) -- Function: int gsl_sf_bessel_y1_e (double X, gsl_sf_result * RESULT) These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x. -- Function: double gsl_sf_bessel_y2 (double X) -- Function: int gsl_sf_bessel_y2_e (double X, gsl_sf_result * RESULT) These routines compute the irregular spherical Bessel function of second order, y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x). -- Function: double gsl_sf_bessel_yl (int L, double X) -- Function: int gsl_sf_bessel_yl_e (int L, double X, gsl_sf_result * RESULT) These routines compute the irregular spherical Bessel function of order L, y_l(x), for l >= 0. -- Function: int gsl_sf_bessel_yl_array (int LMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the irregular spherical Bessel functions y_l(x) for l from 0 to LMAX inclusive for lmax >= 0, storing the results in the array RESULT_ARRAY. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.  File: gsl-ref.info, Node: Regular Modified Spherical Bessel Functions, Next: Irregular Modified Spherical Bessel Functions, Prev: Irregular Spherical Bessel Functions, Up: Bessel Functions 7.5.7 Regular Modified Spherical Bessel Functions ------------------------------------------------- The regular modified spherical Bessel functions i_l(x) are related to the modified Bessel functions of fractional order, i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x) -- Function: double gsl_sf_bessel_i0_scaled (double X) -- Function: int gsl_sf_bessel_i0_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified spherical Bessel function of zeroth order, \exp(-|x|) i_0(x). -- Function: double gsl_sf_bessel_i1_scaled (double X) -- Function: int gsl_sf_bessel_i1_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified spherical Bessel function of first order, \exp(-|x|) i_1(x). -- Function: double gsl_sf_bessel_i2_scaled (double X) -- Function: int gsl_sf_bessel_i2_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified spherical Bessel function of second order, \exp(-|x|) i_2(x) -- Function: double gsl_sf_bessel_il_scaled (int L, double X) -- Function: int gsl_sf_bessel_il_scaled_e (int L, double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified spherical Bessel function of order L, \exp(-|x|) i_l(x) -- Function: int gsl_sf_bessel_il_scaled_array (int LMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the scaled regular modified cylindrical Bessel functions \exp(-|x|) i_l(x) for l from 0 to LMAX inclusive for lmax >= 0, storing the results in the array RESULT_ARRAY. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.  File: gsl-ref.info, Node: Irregular Modified Spherical Bessel Functions, Next: Regular Bessel Function - Fractional Order, Prev: Regular Modified Spherical Bessel Functions, Up: Bessel Functions 7.5.8 Irregular Modified Spherical Bessel Functions --------------------------------------------------- The irregular modified spherical Bessel functions k_l(x) are related to the irregular modified Bessel functions of fractional order, k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x). -- Function: double gsl_sf_bessel_k0_scaled (double X) -- Function: int gsl_sf_bessel_k0_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified spherical Bessel function of zeroth order, \exp(x) k_0(x), for x>0. -- Function: double gsl_sf_bessel_k1_scaled (double X) -- Function: int gsl_sf_bessel_k1_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified spherical Bessel function of first order, \exp(x) k_1(x), for x>0. -- Function: double gsl_sf_bessel_k2_scaled (double X) -- Function: int gsl_sf_bessel_k2_scaled_e (double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified spherical Bessel function of second order, \exp(x) k_2(x), for x>0. -- Function: double gsl_sf_bessel_kl_scaled (int L, double X) -- Function: int gsl_sf_bessel_kl_scaled_e (int L, double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified spherical Bessel function of order L, \exp(x) k_l(x), for x>0. -- Function: int gsl_sf_bessel_kl_scaled_array (int LMAX, double X, double RESULT_ARRAY[]) This routine computes the values of the scaled irregular modified spherical Bessel functions \exp(x) k_l(x) for l from 0 to LMAX inclusive for lmax >= 0 and x>0, storing the results in the array RESULT_ARRAY. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.  File: gsl-ref.info, Node: Regular Bessel Function - Fractional Order, Next: Irregular Bessel Functions - Fractional Order, Prev: Irregular Modified Spherical Bessel Functions, Up: Bessel Functions 7.5.9 Regular Bessel Function--Fractional Order ----------------------------------------------- -- Function: double gsl_sf_bessel_Jnu (double NU, double X) -- Function: int gsl_sf_bessel_Jnu_e (double NU, double X, gsl_sf_result * RESULT) These routines compute the regular cylindrical Bessel function of fractional order \nu, J_\nu(x). -- Function: int gsl_sf_bessel_sequence_Jnu_e (double NU, gsl_mode_t MODE, size_t SIZE, double V[]) This function computes the regular cylindrical Bessel function of fractional order \nu, J_\nu(x), evaluated at a series of x values. The array V of length SIZE contains the x values. They are assumed to be strictly ordered and positive. The array is over-written with the values of J_\nu(x_i).  File: gsl-ref.info, Node: Irregular Bessel Functions - Fractional Order, Next: Regular Modified Bessel Functions - Fractional Order, Prev: Regular Bessel Function - Fractional Order, Up: Bessel Functions 7.5.10 Irregular Bessel Functions--Fractional Order --------------------------------------------------- -- Function: double gsl_sf_bessel_Ynu (double NU, double X) -- Function: int gsl_sf_bessel_Ynu_e (double NU, double X, gsl_sf_result * RESULT) These routines compute the irregular cylindrical Bessel function of fractional order \nu, Y_\nu(x).  File: gsl-ref.info, Node: Regular Modified Bessel Functions - Fractional Order, Next: Irregular Modified Bessel Functions - Fractional Order, Prev: Irregular Bessel Functions - Fractional Order, Up: Bessel Functions 7.5.11 Regular Modified Bessel Functions--Fractional Order ---------------------------------------------------------- -- Function: double gsl_sf_bessel_Inu (double NU, double X) -- Function: int gsl_sf_bessel_Inu_e (double NU, double X, gsl_sf_result * RESULT) These routines compute the regular modified Bessel function of fractional order \nu, I_\nu(x) for x>0, \nu>0. -- Function: double gsl_sf_bessel_Inu_scaled (double NU, double X) -- Function: int gsl_sf_bessel_Inu_scaled_e (double NU, double X, gsl_sf_result * RESULT) These routines compute the scaled regular modified Bessel function of fractional order \nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.  File: gsl-ref.info, Node: Irregular Modified Bessel Functions - Fractional Order, Next: Zeros of Regular Bessel Functions, Prev: Regular Modified Bessel Functions - Fractional Order, Up: Bessel Functions 7.5.12 Irregular Modified Bessel Functions--Fractional Order ------------------------------------------------------------ -- Function: double gsl_sf_bessel_Knu (double NU, double X) -- Function: int gsl_sf_bessel_Knu_e (double NU, double X, gsl_sf_result * RESULT) These routines compute the irregular modified Bessel function of fractional order \nu, K_\nu(x) for x>0, \nu>0. -- Function: double gsl_sf_bessel_lnKnu (double NU, double X) -- Function: int gsl_sf_bessel_lnKnu_e (double NU, double X, gsl_sf_result * RESULT) These routines compute the logarithm of the irregular modified Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0, \nu>0. -- Function: double gsl_sf_bessel_Knu_scaled (double NU, double X) -- Function: int gsl_sf_bessel_Knu_scaled_e (double NU, double X, gsl_sf_result * RESULT) These routines compute the scaled irregular modified Bessel function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0, \nu>0.  File: gsl-ref.info, Node: Zeros of Regular Bessel Functions, Prev: Irregular Modified Bessel Functions - Fractional Order, Up: Bessel Functions 7.5.13 Zeros of Regular Bessel Functions ---------------------------------------- -- Function: double gsl_sf_bessel_zero_J0 (unsigned int S) -- Function: int gsl_sf_bessel_zero_J0_e (unsigned int S, gsl_sf_result * RESULT) These routines compute the location of the S-th positive zero of the Bessel function J_0(x). -- Function: double gsl_sf_bessel_zero_J1 (unsigned int S) -- Function: int gsl_sf_bessel_zero_J1_e (unsigned int S, gsl_sf_result * RESULT) These routines compute the location of the S-th positive zero of the Bessel function J_1(x). -- Function: double gsl_sf_bessel_zero_Jnu (double NU, unsigned int S) -- Function: int gsl_sf_bessel_zero_Jnu_e (double NU, unsigned int S, gsl_sf_result * RESULT) These routines compute the location of the S-th positive zero of the Bessel function J_\nu(x). The current implementation does not support negative values of NU.  File: gsl-ref.info, Node: Clausen Functions, Next: Coulomb Functions, Prev: Bessel Functions, Up: Special Functions 7.6 Clausen Functions ===================== The Clausen function is defined by the following integral, Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2)) It is related to the dilogarithm by Cl_2(\theta) = \Im Li_2(\exp(i\theta)). The Clausen functions are declared in the header file `gsl_sf_clausen.h'. -- Function: double gsl_sf_clausen (double X) -- Function: int gsl_sf_clausen_e (double X, gsl_sf_result * RESULT) These routines compute the Clausen integral Cl_2(x).  File: gsl-ref.info, Node: Coulomb Functions, Next: Coupling Coefficients, Prev: Clausen Functions, Up: Special Functions 7.7 Coulomb Functions ===================== The prototypes of the Coulomb functions are declared in the header file `gsl_sf_coulomb.h'. Both bound state and scattering solutions are available. * Menu: * Normalized Hydrogenic Bound States:: * Coulomb Wave Functions:: * Coulomb Wave Function Normalization Constant::  File: gsl-ref.info, Node: Normalized Hydrogenic Bound States, Next: Coulomb Wave Functions, Up: Coulomb Functions 7.7.1 Normalized Hydrogenic Bound States ---------------------------------------- -- Function: double gsl_sf_hydrogenicR_1 (double Z, double R) -- Function: int gsl_sf_hydrogenicR_1_e (double Z, double R, gsl_sf_result * RESULT) These routines compute the lowest-order normalized hydrogenic bound state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r). -- Function: double gsl_sf_hydrogenicR (int N, int L, double Z, double R) -- Function: int gsl_sf_hydrogenicR_e (int N, int L, double Z, double R, gsl_sf_result * RESULT) These routines compute the N-th normalized hydrogenic bound state radial wavefunction, R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l L^{2l+1}_{n-l-1}(2Zr/n). where L^a_b(x) is the generalized Laguerre polynomial (*note Laguerre Functions::). The normalization is chosen such that the wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.  File: gsl-ref.info, Node: Coulomb Wave Functions, Next: Coulomb Wave Function Normalization Constant, Prev: Normalized Hydrogenic Bound States, Up: Coulomb Functions 7.7.2 Coulomb Wave Functions ---------------------------- The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are described in Abramowitz & Stegun, Chapter 14. Because there can be a large dynamic range of values for these functions, overflows are handled gracefully. If an overflow occurs, `GSL_EOVRFLW' is signalled and exponent(s) are returned through the modifiable parameters EXP_F, EXP_G. The full solution can be reconstructed from the following relations, F_L(eta,x) = fc[k_L] * exp(exp_F) G_L(eta,x) = gc[k_L] * exp(exp_G) F_L'(eta,x) = fcp[k_L] * exp(exp_F) G_L'(eta,x) = gcp[k_L] * exp(exp_G) -- Function: int gsl_sf_coulomb_wave_FG_e (double ETA, double X, double L_F, int K, gsl_sf_result * F, gsl_sf_result * FP, gsl_sf_result * G, gsl_sf_result * GP, double * EXP_F, double * EXP_G) This function computes the Coulomb wave functions F_L(\eta,x), G_{L-k}(\eta,x) and their derivatives F'_L(\eta,x), G'_{L-k}(\eta,x) with respect to x. The parameters are restricted to L, L-k > -1/2, x > 0 and integer k. Note that L itself is not restricted to being an integer. The results are stored in the parameters F, G for the function values and FP, GP for the derivative values. If an overflow occurs, `GSL_EOVRFLW' is returned and scaling exponents are stored in the modifiable parameters EXP_F, EXP_G. -- Function: int gsl_sf_coulomb_wave_F_array (double L_MIN, int KMAX, double ETA, double X, double FC_ARRAY[], double * F_EXPONENT) This function computes the Coulomb wave function F_L(\eta,x) for L = Lmin \dots Lmin + kmax, storing the results in FC_ARRAY. In the case of overflow the exponent is stored in F_EXPONENT. -- Function: int gsl_sf_coulomb_wave_FG_array (double L_MIN, int KMAX, double ETA, double X, double FC_ARRAY[], double GC_ARRAY[], double * F_EXPONENT, double * G_EXPONENT) This function computes the functions F_L(\eta,x), G_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in FC_ARRAY and GC_ARRAY. In the case of overflow the exponents are stored in F_EXPONENT and G_EXPONENT. -- Function: int gsl_sf_coulomb_wave_FGp_array (double L_MIN, int KMAX, double ETA, double X, double FC_ARRAY[], double FCP_ARRAY[], double GC_ARRAY[], double GCP_ARRAY[], double * F_EXPONENT, double * G_EXPONENT) This function computes the functions F_L(\eta,x), G_L(\eta,x) and their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in FC_ARRAY, GC_ARRAY, FCP_ARRAY and GCP_ARRAY. In the case of overflow the exponents are stored in F_EXPONENT and G_EXPONENT. -- Function: int gsl_sf_coulomb_wave_sphF_array (double L_MIN, int KMAX, double ETA, double X, double FC_ARRAY[], double F_EXPONENT[]) This function computes the Coulomb wave function divided by the argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the results in FC_ARRAY. In the case of overflow the exponent is stored in F_EXPONENT. This function reduces to spherical Bessel functions in the limit \eta \to 0.  File: gsl-ref.info, Node: Coulomb Wave Function Normalization Constant, Prev: Coulomb Wave Functions, Up: Coulomb Functions 7.7.3 Coulomb Wave Function Normalization Constant -------------------------------------------------- The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7. -- Function: int gsl_sf_coulomb_CL_e (double L, double ETA, gsl_sf_result * RESULT) This function computes the Coulomb wave function normalization constant C_L(\eta) for L > -1. -- Function: int gsl_sf_coulomb_CL_array (double LMIN, int KMAX, double ETA, double CL[]) This function computes the Coulomb wave function normalization constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.  File: gsl-ref.info, Node: Coupling Coefficients, Next: Dawson Function, Prev: Coulomb Functions, Up: Special Functions 7.8 Coupling Coefficients ========================= The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for combined angular momentum vectors. Since the arguments of the standard coupling coefficient functions are integer or half-integer, the arguments of the following functions are, by convention, integers equal to twice the actual spin value. For information on the 3-j coefficients see Abramowitz & Stegun, Section 27.9. The functions described in this section are declared in the header file `gsl_sf_coupling.h'. * Menu: * 3-j Symbols:: * 6-j Symbols:: * 9-j Symbols::  File: gsl-ref.info, Node: 3-j Symbols, Next: 6-j Symbols, Up: Coupling Coefficients 7.8.1 3-j Symbols ----------------- -- Function: double gsl_sf_coupling_3j (int TWO_JA, int TWO_JB, int TWO_JC, int TWO_MA, int TWO_MB, int TWO_MC) -- Function: int gsl_sf_coupling_3j_e (int TWO_JA, int TWO_JB, int TWO_JC, int TWO_MA, int TWO_MB, int TWO_MC, gsl_sf_result * RESULT) These routines compute the Wigner 3-j coefficient, (ja jb jc ma mb mc) where the arguments are given in half-integer units, ja = TWO_JA/2, ma = TWO_MA/2, etc.  File: gsl-ref.info, Node: 6-j Symbols, Next: 9-j Symbols, Prev: 3-j Symbols, Up: Coupling Coefficients 7.8.2 6-j Symbols ----------------- -- Function: double gsl_sf_coupling_6j (int TWO_JA, int TWO_JB, int TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF) -- Function: int gsl_sf_coupling_6j_e (int TWO_JA, int TWO_JB, int TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, gsl_sf_result * RESULT) These routines compute the Wigner 6-j coefficient, {ja jb jc jd je jf} where the arguments are given in half-integer units, ja = TWO_JA/2, ma = TWO_MA/2, etc.  File: gsl-ref.info, Node: 9-j Symbols, Prev: 6-j Symbols, Up: Coupling Coefficients 7.8.3 9-j Symbols ----------------- -- Function: double gsl_sf_coupling_9j (int TWO_JA, int TWO_JB, int TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, int TWO_JG, int TWO_JH, int TWO_JI) -- Function: int gsl_sf_coupling_9j_e (int TWO_JA, int TWO_JB, int TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, int TWO_JG, int TWO_JH, int TWO_JI, gsl_sf_result * RESULT) These routines compute the Wigner 9-j coefficient, {ja jb jc jd je jf jg jh ji} where the arguments are given in half-integer units, ja = TWO_JA/2, ma = TWO_MA/2, etc.  File: gsl-ref.info, Node: Dawson Function, Next: Debye Functions, Prev: Coupling Coefficients, Up: Special Functions 7.9 Dawson Function =================== The Dawson integral is defined by \exp(-x^2) \int_0^x dt \exp(t^2). A table of Dawson's integral can be found in Abramowitz & Stegun, Table 7.5. The Dawson functions are declared in the header file `gsl_sf_dawson.h'. -- Function: double gsl_sf_dawson (double X) -- Function: int gsl_sf_dawson_e (double X, gsl_sf_result * RESULT) These routines compute the value of Dawson's integral for X.  File: gsl-ref.info, Node: Debye Functions, Next: Dilogarithm, Prev: Dawson Function, Up: Special Functions 7.10 Debye Functions ==================== The Debye functions D_n(x) are defined by the following integral, D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1)) For further information see Abramowitz & Stegun, Section 27.1. The Debye functions are declared in the header file `gsl_sf_debye.h'. -- Function: double gsl_sf_debye_1 (double X) -- Function: int gsl_sf_debye_1_e (double X, gsl_sf_result * RESULT) These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)). -- Function: double gsl_sf_debye_2 (double X) -- Function: int gsl_sf_debye_2_e (double X, gsl_sf_result * RESULT) These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)). -- Function: double gsl_sf_debye_3 (double X) -- Function: int gsl_sf_debye_3_e (double X, gsl_sf_result * RESULT) These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)). -- Function: double gsl_sf_debye_4 (double X) -- Function: int gsl_sf_debye_4_e (double X, gsl_sf_result * RESULT) These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)). -- Function: double gsl_sf_debye_5 (double X) -- Function: int gsl_sf_debye_5_e (double X, gsl_sf_result * RESULT) These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)). -- Function: double gsl_sf_debye_6 (double X) -- Function: int gsl_sf_debye_6_e (double X, gsl_sf_result * RESULT) These routines compute the sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)).  File: gsl-ref.info, Node: Dilogarithm, Next: Elementary Operations, Prev: Debye Functions, Up: Special Functions 7.11 Dilogarithm ================ The functions described in this section are declared in the header file `gsl_sf_dilog.h'. * Menu: * Real Argument:: * Complex Argument::  File: gsl-ref.info, Node: Real Argument, Next: Complex Argument, Up: Dilogarithm 7.11.1 Real Argument -------------------- -- Function: double gsl_sf_dilog (double X) -- Function: int gsl_sf_dilog_e (double X, gsl_sf_result * RESULT) These routines compute the dilogarithm for a real argument. In Lewin's notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for x <= 1, and -\pi\log(x) for x > 1.  File: gsl-ref.info, Node: Complex Argument, Prev: Real Argument, Up: Dilogarithm 7.11.2 Complex Argument ----------------------- -- Function: int gsl_sf_complex_dilog_e (double R, double THETA, gsl_sf_result * RESULT_RE, gsl_sf_result * RESULT_IM) This function computes the full complex-valued dilogarithm for the complex argument z = r \exp(i \theta). The real and imaginary parts of the result are returned in RESULT_RE, RESULT_IM.  File: gsl-ref.info, Node: Elementary Operations, Next: Elliptic Integrals, Prev: Dilogarithm, Up: Special Functions 7.12 Elementary Operations ========================== The following functions allow for the propagation of errors when combining quantities by multiplication. The functions are declared in the header file `gsl_sf_elementary.h'. -- Function: int gsl_sf_multiply_e (double X, double Y, gsl_sf_result * RESULT) This function multiplies X and Y storing the product and its associated error in RESULT. -- Function: int gsl_sf_multiply_err_e (double X, double DX, double Y, double DY, gsl_sf_result * RESULT) This function multiplies X and Y with associated absolute errors DX and DY. The product xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2) is stored in RESULT.  File: gsl-ref.info, Node: Elliptic Integrals, Next: Elliptic Functions (Jacobi), Prev: Elementary Operations, Up: Special Functions 7.13 Elliptic Integrals ======================= The functions described in this section are declared in the header file `gsl_sf_ellint.h'. Further information about the elliptic integrals can be found in Abramowitz & Stegun, Chapter 17. * Menu: * Definition of Legendre Forms:: * Definition of Carlson Forms:: * Legendre Form of Complete Elliptic Integrals:: * Legendre Form of Incomplete Elliptic Integrals:: * Carlson Forms::  File: gsl-ref.info, Node: Definition of Legendre Forms, Next: Definition of Carlson Forms, Up: Elliptic Integrals 7.13.1 Definition of Legendre Forms ----------------------------------- The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and \Pi(\phi,k,n) are defined by, F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t))) E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t))) Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t))) The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) = E(\pi/2, k). The notation used here is based on Carlson, `Numerische Mathematik' 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun, where the functions are given in terms of the parameter m = k^2 and n is replaced by -n.  File: gsl-ref.info, Node: Definition of Carlson Forms, Next: Legendre Form of Complete Elliptic Integrals, Prev: Definition of Legendre Forms, Up: Elliptic Integrals 7.13.2 Definition of Carlson Forms ---------------------------------- The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by, RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1) RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2) RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) RJ(x,y,z,p) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)  File: gsl-ref.info, Node: Legendre Form of Complete Elliptic Integrals, Next: Legendre Form of Incomplete Elliptic Integrals, Prev: Definition of Carlson Forms, Up: Elliptic Integrals 7.13.3 Legendre Form of Complete Elliptic Integrals --------------------------------------------------- -- Function: double gsl_sf_ellint_Kcomp (double K, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_Kcomp_e (double K, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable MODE. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. -- Function: double gsl_sf_ellint_Ecomp (double K, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_Ecomp_e (double K, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable MODE. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. -- Function: double gsl_sf_ellint_Pcomp (double K, double N, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_Pcomp_e (double K, double N, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the complete elliptic integral \Pi(k,n) to the accuracy specified by the mode variable MODE. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.  File: gsl-ref.info, Node: Legendre Form of Incomplete Elliptic Integrals, Next: Carlson Forms, Prev: Legendre Form of Complete Elliptic Integrals, Up: Elliptic Integrals 7.13.4 Legendre Form of Incomplete Elliptic Integrals ----------------------------------------------------- -- Function: double gsl_sf_ellint_F (double PHI, double K, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_F_e (double PHI, double K, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the incomplete elliptic integral F(\phi,k) to the accuracy specified by the mode variable MODE. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. -- Function: double gsl_sf_ellint_E (double PHI, double K, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_E_e (double PHI, double K, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the incomplete elliptic integral E(\phi,k) to the accuracy specified by the mode variable MODE. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. -- Function: double gsl_sf_ellint_P (double PHI, double K, double N, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_P_e (double PHI, double K, double N, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the incomplete elliptic integral \Pi(\phi,k,n) to the accuracy specified by the mode variable MODE. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n. -- Function: double gsl_sf_ellint_D (double PHI, double K, double N, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_D_e (double PHI, double K, double N, gsl_mode_t MODE, gsl_sf_result * RESULT) These functions compute the incomplete elliptic integral D(\phi,k) which is defined through the Carlson form RD(x,y,z) by the following relation, D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1). The argument N is not used and will be removed in a future release.  File: gsl-ref.info, Node: Carlson Forms, Prev: Legendre Form of Incomplete Elliptic Integrals, Up: Elliptic Integrals 7.13.5 Carlson Forms -------------------- -- Function: double gsl_sf_ellint_RC (double X, double Y, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_RC_e (double X, double Y, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the incomplete elliptic integral RC(x,y) to the accuracy specified by the mode variable MODE. -- Function: double gsl_sf_ellint_RD (double X, double Y, double Z, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_RD_e (double X, double Y, double Z, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the incomplete elliptic integral RD(x,y,z) to the accuracy specified by the mode variable MODE. -- Function: double gsl_sf_ellint_RF (double X, double Y, double Z, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_RF_e (double X, double Y, double Z, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the incomplete elliptic integral RF(x,y,z) to the accuracy specified by the mode variable MODE. -- Function: double gsl_sf_ellint_RJ (double X, double Y, double Z, double P, gsl_mode_t MODE) -- Function: int gsl_sf_ellint_RJ_e (double X, double Y, double Z, double P, gsl_mode_t MODE, gsl_sf_result * RESULT) These routines compute the incomplete elliptic integral RJ(x,y,z,p) to the accuracy specified by the mode variable MODE.  File: gsl-ref.info, Node: Elliptic Functions (Jacobi), Next: Error Functions, Prev: Elliptic Integrals, Up: Special Functions 7.14 Elliptic Functions (Jacobi) ================================ The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16. The functions are declared in the header file `gsl_sf_elljac.h'. -- Function: int gsl_sf_elljac_e (double U, double M, double * SN, double * CN, double * DN) This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations.  File: gsl-ref.info, Node: Error Functions, Next: Exponential Functions, Prev: Elliptic Functions (Jacobi), Up: Special Functions 7.15 Error Functions ==================== The error function is described in Abramowitz & Stegun, Chapter 7. The functions in this section are declared in the header file `gsl_sf_erf.h'. * Menu: * Error Function:: * Complementary Error Function:: * Log Complementary Error Function:: * Probability functions::  File: gsl-ref.info, Node: Error Function, Next: Complementary Error Function, Up: Error Functions 7.15.1 Error Function --------------------- -- Function: double gsl_sf_erf (double X) -- Function: int gsl_sf_erf_e (double X, gsl_sf_result * RESULT) These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).  File: gsl-ref.info, Node: Complementary Error Function, Next: Log Complementary Error Function, Prev: Error Function, Up: Error Functions 7.15.2 Complementary Error Function ----------------------------------- -- Function: double gsl_sf_erfc (double X) -- Function: int gsl_sf_erfc_e (double X, gsl_sf_result * RESULT) These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).  File: gsl-ref.info, Node: Log Complementary Error Function, Next: Probability functions, Prev: Complementary Error Function, Up: Error Functions 7.15.3 Log Complementary Error Function --------------------------------------- -- Function: double gsl_sf_log_erfc (double X) -- Function: int gsl_sf_log_erfc_e (double X, gsl_sf_result * RESULT) These routines compute the logarithm of the complementary error function \log(\erfc(x)).  File: gsl-ref.info, Node: Probability functions, Prev: Log Complementary Error Function, Up: Error Functions 7.15.4 Probability functions ---------------------------- The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2. -- Function: double gsl_sf_erf_Z (double X) -- Function: int gsl_sf_erf_Z_e (double X, gsl_sf_result * RESULT) These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2). -- Function: double gsl_sf_erf_Q (double X) -- Function: int gsl_sf_erf_Q_e (double X, gsl_sf_result * RESULT) These routines compute the upper tail of the Gaussian probability function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2). The "hazard function" for the normal distribution, also known as the inverse Mill's ratio, is defined as, h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2) It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim x as x approaches +\infty. -- Function: double gsl_sf_hazard (double X) -- Function: int gsl_sf_hazard_e (double X, gsl_sf_result * RESULT) These routines compute the hazard function for the normal distribution.  File: gsl-ref.info, Node: Exponential Functions, Next: Exponential Integrals, Prev: Error Functions, Up: Special Functions 7.16 Exponential Functions ========================== The functions described in this section are declared in the header file `gsl_sf_exp.h'. * Menu: * Exponential Function:: * Relative Exponential Functions:: * Exponentiation With Error Estimate::  File: gsl-ref.info, Node: Exponential Function, Next: Relative Exponential Functions, Up: Exponential Functions 7.16.1 Exponential Function --------------------------- -- Function: double gsl_sf_exp (double X) -- Function: int gsl_sf_exp_e (double X, gsl_sf_result * RESULT) These routines provide an exponential function \exp(x) using GSL semantics and error checking. -- Function: int gsl_sf_exp_e10_e (double X, gsl_sf_result_e10 * RESULT) This function computes the exponential \exp(x) using the `gsl_sf_result_e10' type to return a result with extended range. This function may be useful if the value of \exp(x) would overflow the numeric range of `double'. -- Function: double gsl_sf_exp_mult (double X, double Y) -- Function: int gsl_sf_exp_mult_e (double X, double Y, gsl_sf_result * RESULT) These routines exponentiate X and multiply by the factor Y to return the product y \exp(x). -- Function: int gsl_sf_exp_mult_e10_e (const double X, const double Y, gsl_sf_result_e10 * RESULT) This function computes the product y \exp(x) using the `gsl_sf_result_e10' type to return a result with extended numeric range.  File: gsl-ref.info, Node: Relative Exponential Functions, Next: Exponentiation With Error Estimate, Prev: Exponential Function, Up: Exponential Functions 7.16.2 Relative Exponential Functions ------------------------------------- -- Function: double gsl_sf_expm1 (double X) -- Function: int gsl_sf_expm1_e (double X, gsl_sf_result * RESULT) These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x. -- Function: double gsl_sf_exprel (double X) -- Function: int gsl_sf_exprel_e (double X, gsl_sf_result * RESULT) These routines compute the quantity (\exp(x)-1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) + x^3/(2*3*4) + \dots. -- Function: double gsl_sf_exprel_2 (double X) -- Function: int gsl_sf_exprel_2_e (double X, gsl_sf_result * RESULT) These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots. -- Function: double gsl_sf_exprel_n (int N, double X) -- Function: int gsl_sf_exprel_n_e (int N, double X, gsl_sf_result * RESULT) These routines compute the N-relative exponential, which is the N-th generalization of the functions `gsl_sf_exprel' and `gsl_sf_exprel2'. The N-relative exponential is given by, exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!) = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ... = 1F1 (1,1+N,x)  File: gsl-ref.info, Node: Exponentiation With Error Estimate, Prev: Relative Exponential Functions, Up: Exponential Functions 7.16.3 Exponentiation With Error Estimate ----------------------------------------- -- Function: int gsl_sf_exp_err_e (double X, double DX, gsl_sf_result * RESULT) This function exponentiates X with an associated absolute error DX. -- Function: int gsl_sf_exp_err_e10_e (double X, double DX, gsl_sf_result_e10 * RESULT) This function exponentiates a quantity X with an associated absolute error DX using the `gsl_sf_result_e10' type to return a result with extended range. -- Function: int gsl_sf_exp_mult_err_e (double X, double DX, double Y, double DY, gsl_sf_result * RESULT) This routine computes the product y \exp(x) for the quantities X, Y with associated absolute errors DX, DY. -- Function: int gsl_sf_exp_mult_err_e10_e (double X, double DX, double Y, double DY, gsl_sf_result_e10 * RESULT) This routine computes the product y \exp(x) for the quantities X, Y with associated absolute errors DX, DY using the `gsl_sf_result_e10' type to return a result with extended range.  File: gsl-ref.info, Node: Exponential Integrals, Next: Fermi-Dirac Function, Prev: Exponential Functions, Up: Special Functions 7.17 Exponential Integrals ========================== Information on the exponential integrals can be found in Abramowitz & Stegun, Chapter 5. These functions are declared in the header file `gsl_sf_expint.h'. * Menu: * Exponential Integral:: * Ei(x):: * Hyperbolic Integrals:: * Ei_3(x):: * Trigonometric Integrals:: * Arctangent Integral::  File: gsl-ref.info, Node: Exponential Integral, Next: Ei(x), Up: Exponential Integrals 7.17.1 Exponential Integral --------------------------- -- Function: double gsl_sf_expint_E1 (double X) -- Function: int gsl_sf_expint_E1_e (double X, gsl_sf_result * RESULT) These routines compute the exponential integral E_1(x), E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t. -- Function: double gsl_sf_expint_E2 (double X) -- Function: int gsl_sf_expint_E2_e (double X, gsl_sf_result * RESULT) These routines compute the second-order exponential integral E_2(x), E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.  File: gsl-ref.info, Node: Ei(x), Next: Hyperbolic Integrals, Prev: Exponential Integral, Up: Exponential Integrals 7.17.2 Ei(x) ------------ -- Function: double gsl_sf_expint_Ei (double X) -- Function: int gsl_sf_expint_Ei_e (double X, gsl_sf_result * RESULT) These routines compute the exponential integral Ei(x), Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t) where PV denotes the principal value of the integral.  File: gsl-ref.info, Node: Hyperbolic Integrals, Next: Ei_3(x), Prev: Ei(x), Up: Exponential Integrals 7.17.3 Hyperbolic Integrals --------------------------- -- Function: double gsl_sf_Shi (double X) -- Function: int gsl_sf_Shi_e (double X, gsl_sf_result * RESULT) These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t. -- Function: double gsl_sf_Chi (double X) -- Function: int gsl_sf_Chi_e (double X, gsl_sf_result * RESULT) These routines compute the integral Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t] , where \gamma_E is the Euler constant (available as the macro `M_EULER').  File: gsl-ref.info, Node: Ei_3(x), Next: Trigonometric Integrals, Prev: Hyperbolic Integrals, Up: Exponential Integrals 7.17.4 Ei_3(x) -------------- -- Function: double gsl_sf_expint_3 (double X) -- Function: int gsl_sf_expint_3_e (double X, gsl_sf_result * RESULT) These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.  File: gsl-ref.info, Node: Trigonometric Integrals, Next: Arctangent Integral, Prev: Ei_3(x), Up: Exponential Integrals 7.17.5 Trigonometric Integrals ------------------------------ -- Function: double gsl_sf_Si (const double X) -- Function: int gsl_sf_Si_e (double X, gsl_sf_result * RESULT) These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t. -- Function: double gsl_sf_Ci (const double X) -- Function: int gsl_sf_Ci_e (double X, gsl_sf_result * RESULT) These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.  File: gsl-ref.info, Node: Arctangent Integral, Prev: Trigonometric Integrals, Up: Exponential Integrals 7.17.6 Arctangent Integral -------------------------- -- Function: double gsl_sf_atanint (double X) -- Function: int gsl_sf_atanint_e (double X, gsl_sf_result * RESULT) These routines compute the Arctangent integral, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.  File: gsl-ref.info, Node: Fermi-Dirac Function, Next: Gamma and Beta Functions, Prev: Exponential Integrals, Up: Special Functions 7.18 Fermi-Dirac Function ========================= The functions described in this section are declared in the header file `gsl_sf_fermi_dirac.h'. * Menu: * Complete Fermi-Dirac Integrals:: * Incomplete Fermi-Dirac Integrals::  File: gsl-ref.info, Node: Complete Fermi-Dirac Integrals, Next: Incomplete Fermi-Dirac Integrals, Up: Fermi-Dirac Function 7.18.1 Complete Fermi-Dirac Integrals ------------------------------------- The complete Fermi-Dirac integral F_j(x) is given by, F_j(x) := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1)) -- Function: double gsl_sf_fermi_dirac_m1 (double X) -- Function: int gsl_sf_fermi_dirac_m1_e (double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral with an index of -1. This integral is given by F_{-1}(x) = e^x / (1 + e^x). -- Function: double gsl_sf_fermi_dirac_0 (double X) -- Function: int gsl_sf_fermi_dirac_0_e (double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral with an index of 0. This integral is given by F_0(x) = \ln(1 + e^x). -- Function: double gsl_sf_fermi_dirac_1 (double X) -- Function: int gsl_sf_fermi_dirac_1_e (double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral with an index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)). -- Function: double gsl_sf_fermi_dirac_2 (double X) -- Function: int gsl_sf_fermi_dirac_2_e (double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral with an index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)). -- Function: double gsl_sf_fermi_dirac_int (int J, double X) -- Function: int gsl_sf_fermi_dirac_int_e (int J, double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral with an integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)). -- Function: double gsl_sf_fermi_dirac_mhalf (double X) -- Function: int gsl_sf_fermi_dirac_mhalf_e (double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral F_{-1/2}(x). -- Function: double gsl_sf_fermi_dirac_half (double X) -- Function: int gsl_sf_fermi_dirac_half_e (double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral F_{1/2}(x). -- Function: double gsl_sf_fermi_dirac_3half (double X) -- Function: int gsl_sf_fermi_dirac_3half_e (double X, gsl_sf_result * RESULT) These routines compute the complete Fermi-Dirac integral F_{3/2}(x).  File: gsl-ref.info, Node: Incomplete Fermi-Dirac Integrals, Prev: Complete Fermi-Dirac Integrals, Up: Fermi-Dirac Function 7.18.2 Incomplete Fermi-Dirac Integrals --------------------------------------- The incomplete Fermi-Dirac integral F_j(x,b) is given by, F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1)) -- Function: double gsl_sf_fermi_dirac_inc_0 (double X, double B) -- Function: int gsl_sf_fermi_dirac_inc_0_e (double X, double B, gsl_sf_result * RESULT) These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).  File: gsl-ref.info, Node: Gamma and Beta Functions, Next: Gegenbauer Functions, Prev: Fermi-Dirac Function, Up: Special Functions 7.19 Gamma and Beta Functions ============================= The functions described in this section are declared in the header file `gsl_sf_gamma.h'. * Menu: * Gamma Functions:: * Factorials:: * Pochhammer Symbol:: * Incomplete Gamma Functions:: * Beta Functions:: * Incomplete Beta Function::  File: gsl-ref.info, Node: Gamma Functions, Next: Factorials, Up: Gamma and Beta Functions 7.19.1 Gamma Functions ---------------------- The Gamma function is defined by the following integral, \Gamma(x) = \int_0^\infty dt t^{x-1} \exp(-t) It is related to the factorial function by \Gamma(n)=(n-1)! for positive integer n. Further information on the Gamma function can be found in Abramowitz & Stegun, Chapter 6. The functions described in this section are declared in the header file `gsl_sf_gamma.h'. -- Function: double gsl_sf_gamma (double X) -- Function: int gsl_sf_gamma_e (double X, gsl_sf_result * RESULT) These routines compute the Gamma function \Gamma(x), subject to x not being a negative integer 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 can be reconstructed using the relation \Gamma(x) = sgn * \exp(resultlg). -- Function: double gsl_sf_gammastar (double X) -- Function: int gsl_sf_gammastar_e (double X, gsl_sf_result * RESULT) These routines compute the regulated Gamma Function \Gamma^*(x) for x > 0. The regulated gamma function is given by, \Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x)) = (1 + (1/12x) + ...) for x \to \infty and is a useful suggestion of Temme. -- Function: double gsl_sf_gammainv (double X) -- Function: int gsl_sf_gammainv_e (double X, gsl_sf_result * RESULT) These routines compute the reciprocal of the gamma function, 1/\Gamma(x) using the real Lanczos method. -- Function: int gsl_sf_lngamma_complex_e (double ZR, double ZI, gsl_sf_result * LNR, gsl_sf_result * ARG) This routine computes \log(\Gamma(z)) for complex z=z_r+i z_i and z not a negative integer or zero, using the complex Lanczos method. The returned parameters are lnr = \log|\Gamma(z)| and arg = \arg(\Gamma(z)) in (-\pi,\pi]. Note that the phase part (ARG) is not well-determined when |z| is very large, due to inevitable roundoff in restricting to (-\pi,\pi]. This will result in a `GSL_ELOSS' error when it occurs. The absolute value part (LNR), however, never suffers from loss of precision.  File: gsl-ref.info, Node: Factorials, Next: Pochhammer Symbol, Prev: Gamma Functions, Up: Gamma and Beta Functions 7.19.2 Factorials ----------------- Although factorials can be computed from the Gamma function, using the relation n! = \Gamma(n+1) for non-negative integer n, it is usually more efficient to call the functions in this section, particularly for small values of n, whose factorial values are maintained in hardcoded tables. -- Function: double gsl_sf_fact (unsigned int N) -- Function: int gsl_sf_fact_e (unsigned int N, gsl_sf_result * RESULT) These routines compute the factorial n!. The factorial is related to the Gamma function by n! = \Gamma(n+1). The maximum value of n such that n! is not considered an overflow is given by the macro `GSL_SF_FACT_NMAX' and is 170. -- Function: double gsl_sf_doublefact (unsigned int N) -- Function: int gsl_sf_doublefact_e (unsigned int N, gsl_sf_result * RESULT) These routines compute the double factorial n!! = n(n-2)(n-4) \dots. The maximum value of n such that n!! is not considered an overflow is given by the macro `GSL_SF_DOUBLEFACT_NMAX' and is 297. -- Function: double gsl_sf_lnfact (unsigned int N) -- Function: int gsl_sf_lnfact_e (unsigned int N, gsl_sf_result * RESULT) These routines compute the logarithm of the factorial of N, \log(n!). The algorithm is faster than computing \ln(\Gamma(n+1)) via `gsl_sf_lngamma' for n < 170, but defers for larger N. -- Function: double gsl_sf_lndoublefact (unsigned int N) -- Function: int gsl_sf_lndoublefact_e (unsigned int N, gsl_sf_result * RESULT) These routines compute the logarithm of the double factorial of N, \log(n!!). -- Function: double gsl_sf_choose (unsigned int N, unsigned int M) -- Function: int gsl_sf_choose_e (unsigned int N, unsigned int M, gsl_sf_result * RESULT) These routines compute the combinatorial factor `n choose m' = n!/(m!(n-m)!) -- Function: double gsl_sf_lnchoose (unsigned int N, unsigned int M) -- Function: int gsl_sf_lnchoose_e (unsigned int N, unsigned int M, gsl_sf_result * RESULT) These routines compute the logarithm of `n choose m'. This is equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!). -- Function: double gsl_sf_taylorcoeff (int N, double X) -- Function: int gsl_sf_taylorcoeff_e (int N, double X, gsl_sf_result * RESULT) These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.  File: gsl-ref.info, Node: Pochhammer Symbol, Next: Incomplete Gamma Functions, Prev: Factorials, Up: Gamma and Beta Functions 7.19.3 Pochhammer Symbol ------------------------ -- Function: double gsl_sf_poch (double A, double X) -- Function: int gsl_sf_poch_e (double A, double X, gsl_sf_result * RESULT) These routines compute the Pochhammer symbol (a)_x = \Gamma(a + x)/\Gamma(a), subject to a and a+x not being negative integers or zero. The Pochhammer symbol is also known as the Apell symbol and sometimes written as (a,x). -- Function: double gsl_sf_lnpoch (double A, double X) -- Function: int gsl_sf_lnpoch_e (double A, double X, gsl_sf_result * RESULT) These routines compute the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)) for a > 0, a+x > 0. -- Function: int gsl_sf_lnpoch_sgn_e (double A, double X, gsl_sf_result * RESULT, double * SGN) These routines compute the sign of the Pochhammer symbol and the logarithm of its magnitude. The computed parameters are result = \log(|(a)_x|) and sgn = \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a), subject to a, a+x not being negative integers or zero. -- Function: double gsl_sf_pochrel (double A, double X) -- Function: int gsl_sf_pochrel_e (double A, double X, gsl_sf_result * RESULT) These routines compute the relative Pochhammer symbol ((a)_x - 1)/x where (a)_x = \Gamma(a + x)/\Gamma(a).  File: gsl-ref.info, Node: Incomplete Gamma Functions, Next: Beta Functions, Prev: Pochhammer Symbol, Up: Gamma and Beta Functions 7.19.4 Incomplete Gamma Functions --------------------------------- -- Function: double gsl_sf_gamma_inc (double A, double X) -- Function: int gsl_sf_gamma_inc_e (double A, double X, gsl_sf_result * RESULT) These functions compute the unnormalized incomplete Gamma Function \Gamma(a,x) = \int_x^\infty dt t^{a-1} \exp(-t) for a real and x >= 0. -- Function: double gsl_sf_gamma_inc_Q (double A, double X) -- Function: int gsl_sf_gamma_inc_Q_e (double A, double X, gsl_sf_result * RESULT) These routines compute the normalized incomplete Gamma Function Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^{a-1} \exp(-t) for a > 0, x >= 0. -- Function: double gsl_sf_gamma_inc_P (double A, double X) -- Function: int gsl_sf_gamma_inc_P_e (double A, double X, gsl_sf_result * RESULT) These routines compute the complementary normalized incomplete Gamma Function P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt t^{a-1} \exp(-t) for a > 0, x >= 0. Note that Abramowitz & Stegun call P(a,x) the incomplete gamma function (section 6.5).  File: gsl-ref.info, Node: Beta Functions, Next: Incomplete Beta Function, Prev: Incomplete Gamma Functions, Up: Gamma and Beta Functions 7.19.5 Beta Functions --------------------- -- Function: double gsl_sf_beta (double A, double B) -- Function: int gsl_sf_beta_e (double A, double B, gsl_sf_result * RESULT) These routines compute the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being negative integers. -- Function: double gsl_sf_lnbeta (double A, double B) -- Function: int gsl_sf_lnbeta_e (double A, double B, gsl_sf_result * RESULT) These routines compute the logarithm of the Beta Function, \log(B(a,b)) subject to a and b not being negative integers.  File: gsl-ref.info, Node: Incomplete Beta Function, Prev: Beta Functions, Up: Gamma and Beta Functions 7.19.6 Incomplete Beta Function ------------------------------- -- Function: double gsl_sf_beta_inc (double A, double B, double X) -- Function: int gsl_sf_beta_inc_e (double A, double B, double X, gsl_sf_result * RESULT) These routines compute the normalized incomplete Beta function I_x(a,b)=B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt for a > 0, b > 0, and 0 <= x <= 1.  File: gsl-ref.info, Node: Gegenbauer Functions, Next: Hypergeometric Functions, Prev: Gamma and Beta Functions, Up: Special Functions 7.20 Gegenbauer Functions ========================= The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials. The functions described in this section are declared in the header file `gsl_sf_gegenbauer.h'. -- Function: double gsl_sf_gegenpoly_1 (double LAMBDA, double X) -- Function: double gsl_sf_gegenpoly_2 (double LAMBDA, double X) -- Function: double gsl_sf_gegenpoly_3 (double LAMBDA, double X) -- Function: int gsl_sf_gegenpoly_1_e (double LAMBDA, double X, gsl_sf_result * RESULT) -- Function: int gsl_sf_gegenpoly_2_e (double LAMBDA, double X, gsl_sf_result * RESULT) -- Function: int gsl_sf_gegenpoly_3_e (double LAMBDA, double X, gsl_sf_result * RESULT) These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3. -- Function: double gsl_sf_gegenpoly_n (int N, double LAMBDA, double X) -- Function: int gsl_sf_gegenpoly_n_e (int N, double LAMBDA, double X, gsl_sf_result * RESULT) These functions evaluate the Gegenbauer polynomial C^{(\lambda)}_n(x) for a specific value of N, LAMBDA, X subject to \lambda > -1/2, n >= 0. -- Function: int gsl_sf_gegenpoly_array (int NMAX, double LAMBDA, double X, double RESULT_ARRAY[]) This function computes an array of Gegenbauer polynomials C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda > -1/2, nmax >= 0.  File: gsl-ref.info, Node: Hypergeometric Functions, Next: Laguerre Functions, Prev: Gegenbauer Functions, Up: Special Functions 7.21 Hypergeometric Functions ============================= Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15. These functions are declared in the header file `gsl_sf_hyperg.h'. -- Function: double gsl_sf_hyperg_0F1 (double C, double X) -- Function: int gsl_sf_hyperg_0F1_e (double C, double X, gsl_sf_result * RESULT) These routines compute the hypergeometric function 0F1(c,x). -- Function: double gsl_sf_hyperg_1F1_int (int M, int N, double X) -- Function: int gsl_sf_hyperg_1F1_int_e (int M, int N, double X, gsl_sf_result * RESULT) These routines compute the confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters M, N. -- Function: double gsl_sf_hyperg_1F1 (double A, double B, double X) -- Function: int gsl_sf_hyperg_1F1_e (double A, double B, double X, gsl_sf_result * RESULT) These routines compute the confluent hypergeometric function 1F1(a,b,x) = M(a,b,x) for general parameters A, B. -- Function: double gsl_sf_hyperg_U_int (int M, int N, double X) -- Function: int gsl_sf_hyperg_U_int_e (int M, int N, double X, gsl_sf_result * RESULT) These routines compute the confluent hypergeometric function U(m,n,x) for integer parameters M, N. -- Function: int gsl_sf_hyperg_U_int_e10_e (int M, int N, double X, gsl_sf_result_e10 * RESULT) This routine computes the confluent hypergeometric function U(m,n,x) for integer parameters M, N using the `gsl_sf_result_e10' type to return a result with extended range. -- Function: double gsl_sf_hyperg_U (double A, double B, double X) -- Function: int gsl_sf_hyperg_U_e (double A, double B, double X, 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) for |x| < 1. If the arguments (a,b,c,x) are too close to a singularity then the function can return the error code `GSL_EMAXITER' when the series approximation converges too slowly. This occurs in the region of x=1, c - a - b = m for integer m. -- Function: double gsl_sf_hyperg_2F1_conj (double AR, double AI, double C, double X) -- Function: int gsl_sf_hyperg_2F1_conj_e (double AR, double AI, double C, double X, gsl_sf_result * RESULT) These routines compute the Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1. exceptions: -- Function: double gsl_sf_hyperg_2F1_renorm (double A, double B, double C, double X) -- Function: int gsl_sf_hyperg_2F1_renorm_e (double A, double B, double C, double X, gsl_sf_result * RESULT) These routines compute the renormalized Gauss hypergeometric function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1. -- Function: double gsl_sf_hyperg_2F1_conj_renorm (double AR, double AI, double C, double X) -- Function: int gsl_sf_hyperg_2F1_conj_renorm_e (double AR, double AI, double C, double X, gsl_sf_result * RESULT) These routines compute the renormalized Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| < 1. -- Function: double gsl_sf_hyperg_2F0 (double A, double B, double X) -- Function: int gsl_sf_hyperg_2F0_e (double A, double B, double X, gsl_sf_result * RESULT) These routines compute the hypergeometric function 2F0(a,b,x). The series representation is a divergent hypergeometric series. However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)  File: gsl-ref.info, Node: Laguerre Functions, Next: Lambert W Functions, Prev: Hypergeometric Functions, Up: Special Functions 7.22 Laguerre Functions ======================= The generalized Laguerre polynomials are defined in terms of confluent hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x), and are sometimes referred to as the associated Laguerre polynomials. They are related to the plain Laguerre polynomials L_n(x) by L^0_n(x) = L_n(x) and L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x). For more information see Abramowitz & Stegun, Chapter 22. The functions described in this section are declared in the header file `gsl_sf_laguerre.h'. -- Function: double gsl_sf_laguerre_1 (double A, double X) -- Function: double gsl_sf_laguerre_2 (double A, double X) -- Function: double gsl_sf_laguerre_3 (double A, double X) -- Function: int gsl_sf_laguerre_1_e (double A, double X, gsl_sf_result * RESULT) -- Function: int gsl_sf_laguerre_2_e (double A, double X, gsl_sf_result * RESULT) -- Function: int gsl_sf_laguerre_3_e (double A, double X, gsl_sf_result * RESULT) These routines evaluate the generalized Laguerre polynomials L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations. -- Function: double gsl_sf_laguerre_n (const int N, const double A, const double X) -- Function: int gsl_sf_laguerre_n_e (int N, double A, double X, gsl_sf_result * RESULT) These routines evaluate the generalized Laguerre polynomials L^a_n(x) for a > -1, n >= 0.  File: gsl-ref.info, Node: Lambert W Functions, Next: Legendre Functions and Spherical Harmonics, Prev: Laguerre Functions, Up: Special Functions 7.23 Lambert W Functions ======================== Lambert's W functions, W(x), are defined to be solutions of the equation W(x) \exp(W(x)) = x. This function has multiple branches for x < 0; however, it has only two real-valued branches. We define W_0(x) to be the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the other real branch, where W < -1 for x < 0. The Lambert functions are declared in the header file `gsl_sf_lambert.h'. -- Function: double gsl_sf_lambert_W0 (double X) -- Function: int gsl_sf_lambert_W0_e (double X, gsl_sf_result * RESULT) These compute the principal branch of the Lambert W function, W_0(x). -- Function: double gsl_sf_lambert_Wm1 (double X) -- Function: int gsl_sf_lambert_Wm1_e (double X, gsl_sf_result * RESULT) These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).  File: gsl-ref.info, Node: Legendre Functions and Spherical Harmonics, Next: Logarithm and Related Functions, Prev: Lambert W Functions, Up: Special Functions 7.24 Legendre Functions and Spherical Harmonics =============================================== The Legendre Functions and Legendre Polynomials are described in Abramowitz & Stegun, Chapter 8. These functions are declared in the header file `gsl_sf_legendre.h'. * Menu: * Legendre Polynomials:: * Associated Legendre Polynomials and Spherical Harmonics:: * Conical Functions:: * Radial Functions for Hyperbolic Space::  File: gsl-ref.info, Node: Legendre Polynomials, Next: Associated Legendre Polynomials and Spherical Harmonics, Up: Legendre Functions and Spherical Harmonics 7.24.1 Legendre Polynomials --------------------------- -- Function: double gsl_sf_legendre_P1 (double X) -- Function: double gsl_sf_legendre_P2 (double X) -- Function: double gsl_sf_legendre_P3 (double X) -- Function: int gsl_sf_legendre_P1_e (double X, gsl_sf_result * RESULT) -- Function: int gsl_sf_legendre_P2_e (double X, gsl_sf_result * RESULT) -- Function: int gsl_sf_legendre_P3_e (double X, gsl_sf_result * RESULT) These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3. -- Function: double gsl_sf_legendre_Pl (int L, double X) -- Function: int gsl_sf_legendre_Pl_e (int L, double X, gsl_sf_result * RESULT) These functions evaluate the Legendre polynomial P_l(x) for a specific value of L, X subject to l >= 0, |x| <= 1 -- Function: int gsl_sf_legendre_Pl_array (int LMAX, double X, double RESULT_ARRAY[]) -- Function: int gsl_sf_legendre_Pl_deriv_array (int LMAX, double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[]) These functions compute an array of Legendre polynomials P_l(x), and optionally their derivatives dP_l(x)/dx, for l = 0, \dots, lmax, |x| <= 1 -- Function: double gsl_sf_legendre_Q0 (double X) -- Function: int gsl_sf_legendre_Q0_e (double X, gsl_sf_result * RESULT) These routines compute the Legendre function Q_0(x) for x > -1, x != 1. -- Function: double gsl_sf_legendre_Q1 (double X) -- Function: int gsl_sf_legendre_Q1_e (double X, gsl_sf_result * RESULT) These routines compute the Legendre function Q_1(x) for x > -1, x != 1. -- Function: double gsl_sf_legendre_Ql (int L, double X) -- Function: int gsl_sf_legendre_Ql_e (int L, double X, gsl_sf_result * RESULT) These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.  File: gsl-ref.info, Node: Associated Legendre Polynomials and Spherical Harmonics, Next: Conical Functions, Prev: Legendre Polynomials, Up: Legendre Functions and Spherical Harmonics 7.24.2 Associated Legendre Polynomials and Spherical Harmonics -------------------------------------------------------------- The following functions compute the associated Legendre Polynomials P_l^m(x). Note that this function grows combinatorially with l and can overflow for l larger than about 150. There is no trouble for small m, but overflow occurs when m and l are both large. Rather than allow overflows, these functions refuse to calculate P_l^m(x) and return `GSL_EOVRFLW' when they can sense that l and m are too big. If you want to calculate a spherical harmonic, then _do not_ use these functions. Instead use `gsl_sf_legendre_sphPlm' below, which uses a similar recursion, but with the normalized functions. -- Function: double gsl_sf_legendre_Plm (int L, int M, double X) -- Function: int gsl_sf_legendre_Plm_e (int L, int M, double X, gsl_sf_result * RESULT) These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1. -- Function: int gsl_sf_legendre_Plm_array (int LMAX, int M, double X, double RESULT_ARRAY[]) -- Function: int gsl_sf_legendre_Plm_deriv_array (int LMAX, int M, double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[]) These functions compute an array of Legendre polynomials P_l^m(x), and optionally their derivatives dP_l^m(x)/dx, for m >= 0, l = |m|, ..., lmax, |x| <= 1. -- Function: double gsl_sf_legendre_sphPlm (int L, int M, double X) -- Function: int gsl_sf_legendre_sphPlm_e (int L, int M, double X, gsl_sf_result * RESULT) These routines compute the normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics. The parameters must satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the overflows that occur for the standard normalization of P_l^m(x). -- Function: int gsl_sf_legendre_sphPlm_array (int LMAX, int M, double X, double RESULT_ARRAY[]) -- Function: int gsl_sf_legendre_sphPlm_deriv_array (int LMAX, int M, double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[]) These functions compute an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$, and optionally their derivatives, for m >= 0, l = |m|, ..., lmax, |x| <= 1.0 -- Function: int gsl_sf_legendre_array_size (const int LMAX, const int M) This function returns the size of RESULT_ARRAY[] needed for the array versions of P_l^m(x), LMAX - M + 1.  File: gsl-ref.info, Node: Conical Functions, Next: Radial Functions for Hyperbolic Space, Prev: Associated Legendre Polynomials and Spherical Harmonics, Up: Legendre Functions and Spherical Harmonics 7.24.3 Conical Functions ------------------------ The Conical Functions P^\mu_{-(1/2)+i\lambda}(x) and Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section 8.12. -- Function: double gsl_sf_conicalP_half (double LAMBDA, double X) -- Function: int gsl_sf_conicalP_half_e (double LAMBDA, double X, gsl_sf_result * RESULT) These routines compute the irregular Spherical Conical Function P^{1/2}_{-1/2 + i \lambda}(x) for x > -1. -- Function: double gsl_sf_conicalP_mhalf (double LAMBDA, double X) -- Function: int gsl_sf_conicalP_mhalf_e (double LAMBDA, double X, gsl_sf_result * RESULT) These routines compute the regular Spherical Conical Function P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1. -- Function: double gsl_sf_conicalP_0 (double LAMBDA, double X) -- Function: int gsl_sf_conicalP_0_e (double LAMBDA, double X, gsl_sf_result * RESULT) These routines compute the conical function P^0_{-1/2 + i \lambda}(x) for x > -1. -- Function: double gsl_sf_conicalP_1 (double LAMBDA, double X) -- Function: int gsl_sf_conicalP_1_e (double LAMBDA, double X, gsl_sf_result * RESULT) These routines compute the conical function P^1_{-1/2 + i \lambda}(x) for x > -1. -- Function: double gsl_sf_conicalP_sph_reg (int L, double LAMBDA, double X) -- Function: int gsl_sf_conicalP_sph_reg_e (int L, double LAMBDA, double X, gsl_sf_result * RESULT) These routines compute the Regular Spherical Conical Function P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, l >= -1. -- Function: double gsl_sf_conicalP_cyl_reg (int M, double LAMBDA, double X) -- Function: int gsl_sf_conicalP_cyl_reg_e (int M, double LAMBDA, double X, gsl_sf_result * RESULT) These routines compute the Regular Cylindrical Conical Function P^{-m}_{-1/2 + i \lambda}(x) for x > -1, m >= -1.  File: gsl-ref.info, Node: Radial Functions for Hyperbolic Space, Prev: Conical Functions, Up: Legendre Functions and Spherical Harmonics 7.24.4 Radial Functions for Hyperbolic Space -------------------------------------------- The following spherical functions are specializations of Legendre functions which give the regular eigenfunctions of the Laplacian on a 3-dimensional hyperbolic space H3d. Of particular interest is the flat limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed. -- Function: double gsl_sf_legendre_H3d_0 (double LAMBDA, double ETA) -- Function: int gsl_sf_legendre_H3d_0_e (double LAMBDA, double ETA, gsl_sf_result * RESULT) These routines compute the zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta)) for \eta >= 0. In the flat limit this takes the form L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta). -- Function: double gsl_sf_legendre_H3d_1 (double LAMBDA, double ETA) -- Function: int gsl_sf_legendre_H3d_1_e (double LAMBDA, double ETA, gsl_sf_result * RESULT) These routines compute the first radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta)) for \eta >= 0. In the flat limit this takes the form L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta). -- Function: double gsl_sf_legendre_H3d (int L, double LAMBDA, double ETA) -- Function: int gsl_sf_legendre_H3d_e (int L, double LAMBDA, double ETA, gsl_sf_result * RESULT) These routines compute the L-th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space \eta >= 0, l >= 0. In the flat limit this takes the form L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta). -- Function: int gsl_sf_legendre_H3d_array (int LMAX, double LAMBDA, double ETA, double RESULT_ARRAY[]) This function computes an array of radial eigenfunctions L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.  File: gsl-ref.info, Node: Logarithm and Related Functions, Next: Mathieu Functions, Prev: Legendre Functions and Spherical Harmonics, Up: Special Functions 7.25 Logarithm and Related Functions ==================================== Information on the properties of the Logarithm function can be found in Abramowitz & Stegun, Chapter 4. The functions described in this section are declared in the header file `gsl_sf_log.h'. -- Function: double gsl_sf_log (double X) -- Function: int gsl_sf_log_e (double X, gsl_sf_result * RESULT) These routines compute the logarithm of X, \log(x), for x > 0. -- Function: double gsl_sf_log_abs (double X) -- Function: int gsl_sf_log_abs_e (double X, gsl_sf_result * RESULT) These routines compute the logarithm of the magnitude of X, \log(|x|), for x \ne 0. -- Function: int gsl_sf_complex_log_e (double ZR, double ZI, gsl_sf_result * LNR, gsl_sf_result * THETA) This routine computes the complex logarithm of z = z_r + i z_i. The results are returned as LNR, THETA such that \exp(lnr + i \theta) = z_r + i z_i, where \theta lies in the range [-\pi,\pi]. -- Function: double gsl_sf_log_1plusx (double X) -- Function: int gsl_sf_log_1plusx_e (double X, gsl_sf_result * RESULT) These routines compute \log(1 + x) for x > -1 using an algorithm that is accurate for small x. -- Function: double gsl_sf_log_1plusx_mx (double X) -- Function: int gsl_sf_log_1plusx_mx_e (double X, gsl_sf_result * RESULT) These routines compute \log(1 + x) - x for x > -1 using an algorithm that is accurate for small x.  File: gsl-ref.info, Node: Mathieu Functions, Next: Power Function, Prev: Logarithm and Related Functions, Up: Special Functions 7.26 Mathieu Functions ====================== The routines described in this section compute the angular and radial Mathieu functions, and their characteristic values. Mathieu functions are the solutions of the following two differential equations: d^2y/dv^2 + (a - 2q\cos 2v)y = 0 d^2f/du^2 - (a - 2q\cosh 2u)f = 0 The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd periodic solutions of the first equation, which is known as Mathieu's equation. These exist only for the discrete sequence of characteristic values a=a_r(q) (even-periodic) and a=b_r(q) (odd-periodic). The radial Mathieu functions Mc^{(j)}_{r}(z,q), Ms^{(j)}_{r}(z,q) are the solutions of the second equation, which is referred to as Mathieu's modified equation. The radial Mathieu functions of the first, second, third and fourth kind are denoted by the parameter j, which takes the value 1, 2, 3 or 4. For more information on the Mathieu functions, see Abramowitz and Stegun, Chapter 20. These functions are defined in the header file `gsl_sf_mathieu.h'. * Menu: * Mathieu Function Workspace:: * Mathieu Function Characteristic Values:: * Angular Mathieu Functions:: * Radial Mathieu Functions::  File: gsl-ref.info, Node: Mathieu Function Workspace, Next: Mathieu Function Characteristic Values, Up: Mathieu Functions 7.26.1 Mathieu Function Workspace --------------------------------- The Mathieu functions can be computed for a single order or for multiple orders, using array-based routines. The array-based routines require a preallocated workspace. -- Function: gsl_sf_mathieu_workspace * gsl_sf_mathieu_alloc (size_t N, double QMAX) This function returns a workspace for the array versions of the Mathieu routines. The arguments N and QMAX specify the maximum order and q-value of Mathieu functions which can be computed with this workspace. -- Function: void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace *WORK) This function frees the workspace WORK.  File: gsl-ref.info, Node: Mathieu Function Characteristic Values, Next: Angular Mathieu Functions, Prev: Mathieu Function Workspace, Up: Mathieu Functions 7.26.2 Mathieu Function Characteristic Values --------------------------------------------- -- Function: int gsl_sf_mathieu_a (int N, double Q, gsl_sf_result *RESULT) -- Function: int gsl_sf_mathieu_b (int N, double Q, gsl_sf_result *RESULT) These routines compute the characteristic values a_n(q), b_n(q) of the Mathieu functions ce_n(q,x) and se_n(q,x), respectively. -- Function: int gsl_sf_mathieu_a_array (int ORDER_MIN, int ORDER_MAX, double Q, gsl_sf_mathieu_workspace *WORK, double RESULT_ARRAY[]) -- Function: int gsl_sf_mathieu_b_array (int ORDER_MIN, int ORDER_MAX, double Q, gsl_sf_mathieu_workspace *WORK, double RESULT_ARRAY[]) These routines compute a series of Mathieu characteristic values a_n(q), b_n(q) for n from ORDER_MIN to ORDER_MAX inclusive, storing the results in the array RESULT_ARRAY.  File: gsl-ref.info, Node: Angular Mathieu Functions, Next: Radial Mathieu Functions, Prev: Mathieu Function Characteristic Values, Up: Mathieu Functions 7.26.3 Angular Mathieu Functions -------------------------------- -- Function: int gsl_sf_mathieu_ce (int N, double Q, double X, gsl_sf_result *RESULT) -- Function: int gsl_sf_mathieu_se (int N, double Q, double X, gsl_sf_result *RESULT) These routines compute the angular Mathieu functions ce_n(q,x) and se_n(q,x), respectively. -- Function: int gsl_sf_mathieu_ce_array (int NMIN, int NMAX, double Q, double X, gsl_sf_mathieu_workspace *WORK, double RESULT_ARRAY[]) -- Function: int gsl_sf_mathieu_se_array (int NMIN, int NMAX, double Q, double X, gsl_sf_mathieu_workspace *WORK, double RESULT_ARRAY[]) These routines compute a series of the angular Mathieu functions ce_n(q,x) and se_n(q,x) of order n from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY.  File: gsl-ref.info, Node: Radial Mathieu Functions, Prev: Angular Mathieu Functions, Up: Mathieu Functions 7.26.4 Radial Mathieu Functions ------------------------------- -- Function: int gsl_sf_mathieu_Mc (int J, int N, double Q, double X, gsl_sf_result *RESULT) -- Function: int gsl_sf_mathieu_Ms (int J, int N, double Q, double X, gsl_sf_result *RESULT) These routines compute the radial J-th kind Mathieu functions Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order N. The allowed values of J are 1 and 2. The functions for j = 3,4 can be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or Ms_n^{(j)}. -- Function: int gsl_sf_mathieu_Mc_array (int J, int NMIN, int NMAX, double Q, double X, gsl_sf_mathieu_workspace *WORK, double RESULT_ARRAY[]) -- Function: int gsl_sf_mathieu_Ms_array (int J, int NMIN, int NMAX, double Q, double X, gsl_sf_mathieu_workspace *WORK, double RESULT_ARRAY[]) These routines compute a series of the radial Mathieu functions of kind J, with order from NMIN to NMAX inclusive, storing the results in the array RESULT_ARRAY.  File: gsl-ref.info, Node: Power Function, Next: Psi (Digamma) Function, Prev: Mathieu Functions, Up: Special Functions 7.27 Power Function =================== The following functions are equivalent to the function `gsl_pow_int' (*note Small integer powers::) with an error estimate. These functions are declared in the header file `gsl_sf_pow_int.h'. -- Function: double gsl_sf_pow_int (double X, int N) -- Function: int gsl_sf_pow_int_e (double X, int N, gsl_sf_result * RESULT) These routines compute the power x^n for integer N. The power is computed using the minimum number of multiplications. For example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications. For reasons of efficiency, these functions do not check for overflow or underflow conditions. #include /* compute 3.0**12 */ double y = gsl_sf_pow_int(3.0, 12);  File: gsl-ref.info, Node: Psi (Digamma) Function, Next: Synchrotron Functions, Prev: Power Function, Up: Special Functions 7.28 Psi (Digamma) Function =========================== The polygamma functions of order n are defined by \psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x)) where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function. These functions are declared in the header file `gsl_sf_psi.h'. * Menu: * Digamma Function:: * Trigamma Function:: * Polygamma Function::  File: gsl-ref.info, Node: Digamma Function, Next: Trigamma Function, Up: Psi (Digamma) Function 7.28.1 Digamma Function ----------------------- -- Function: double gsl_sf_psi_int (int N) -- Function: int gsl_sf_psi_int_e (int N, gsl_sf_result * RESULT) These routines compute the digamma function \psi(n) for positive integer N. The digamma function is also called the Psi function. -- Function: double gsl_sf_psi (double X) -- Function: int gsl_sf_psi_e (double X, gsl_sf_result * RESULT) These routines compute the digamma function \psi(x) for general x, x \ne 0. -- Function: double gsl_sf_psi_1piy (double Y) -- Function: int gsl_sf_psi_1piy_e (double Y, gsl_sf_result * RESULT) These routines compute the real part of the digamma function on the line 1+i y, \Re[\psi(1 + i y)].  File: gsl-ref.info, Node: Trigamma Function, Next: Polygamma Function, Prev: Digamma Function, Up: Psi (Digamma) Function 7.28.2 Trigamma Function ------------------------ -- Function: double gsl_sf_psi_1_int (int N) -- Function: int gsl_sf_psi_1_int_e (int N, gsl_sf_result * RESULT) These routines compute the Trigamma function \psi'(n) for positive integer n. -- Function: double gsl_sf_psi_1 (double X) -- Function: int gsl_sf_psi_1_e (double X, gsl_sf_result * RESULT) These routines compute the Trigamma function \psi'(x) for general x.  File: gsl-ref.info, Node: Polygamma Function, Prev: Trigamma Function, Up: Psi (Digamma) Function 7.28.3 Polygamma Function ------------------------- -- Function: double gsl_sf_psi_n (int N, double X) -- Function: int gsl_sf_psi_n_e (int N, double X, gsl_sf_result * RESULT) These routines compute the polygamma function \psi^{(n)}(x) for n >= 0, x > 0.  File: gsl-ref.info, Node: Synchrotron Functions, Next: Transport Functions, Prev: Psi (Digamma) Function, Up: Special Functions 7.29 Synchrotron Functions ========================== The functions described in this section are declared in the header file `gsl_sf_synchrotron.h'. -- Function: double gsl_sf_synchrotron_1 (double X) -- Function: int gsl_sf_synchrotron_1_e (double X, gsl_sf_result * RESULT) These routines compute the first synchrotron function x \int_x^\infty dt K_{5/3}(t) for x >= 0. -- Function: double gsl_sf_synchrotron_2 (double X) -- Function: int gsl_sf_synchrotron_2_e (double X, gsl_sf_result * RESULT) These routines compute the second synchrotron function x K_{2/3}(x) for x >= 0.  File: gsl-ref.info, Node: Transport Functions, Next: Trigonometric Functions, Prev: Synchrotron Functions, Up: Special Functions 7.30 Transport Functions ======================== The transport functions J(n,x) are defined by the integral representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2. They are declared in the header file `gsl_sf_transport.h'. -- Function: double gsl_sf_transport_2 (double X) -- Function: int gsl_sf_transport_2_e (double X, gsl_sf_result * RESULT) These routines compute the transport function J(2,x). -- Function: double gsl_sf_transport_3 (double X) -- Function: int gsl_sf_transport_3_e (double X, gsl_sf_result * RESULT) These routines compute the transport function J(3,x). -- Function: double gsl_sf_transport_4 (double X) -- Function: int gsl_sf_transport_4_e (double X, gsl_sf_result * RESULT) These routines compute the transport function J(4,x). -- Function: double gsl_sf_transport_5 (double X) -- Function: int gsl_sf_transport_5_e (double X, gsl_sf_result * RESULT) These routines compute the transport function J(5,x).  File: gsl-ref.info, Node: Trigonometric Functions, Next: Zeta Functions, Prev: Transport Functions, Up: Special Functions 7.31 Trigonometric Functions ============================ The library includes its own trigonometric functions in order to provide consistency across platforms and reliable error estimates. These functions are declared in the header file `gsl_sf_trig.h'. * Menu: * Circular Trigonometric Functions:: * Trigonometric Functions for Complex Arguments:: * Hyperbolic Trigonometric Functions:: * Conversion Functions:: * Restriction Functions:: * Trigonometric Functions With Error Estimates::  File: gsl-ref.info, Node: Circular Trigonometric Functions, Next: Trigonometric Functions for Complex Arguments, Up: Trigonometric Functions 7.31.1 Circular Trigonometric Functions --------------------------------------- -- Function: double gsl_sf_sin (double X) -- Function: int gsl_sf_sin_e (double X, gsl_sf_result * RESULT) These routines compute the sine function \sin(x). -- Function: double gsl_sf_cos (double X) -- Function: int gsl_sf_cos_e (double X, gsl_sf_result * RESULT) These routines compute the cosine function \cos(x). -- Function: double gsl_sf_hypot (double X, double Y) -- Function: int gsl_sf_hypot_e (double X, double Y, gsl_sf_result * RESULT) These routines compute the hypotenuse function \sqrt{x^2 + y^2} avoiding overflow and underflow. -- Function: double gsl_sf_sinc (double X) -- Function: int gsl_sf_sinc_e (double X, gsl_sf_result * RESULT) These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any value of X.  File: gsl-ref.info, Node: Trigonometric Functions for Complex Arguments, Next: Hyperbolic Trigonometric Functions, Prev: Circular Trigonometric Functions, Up: Trigonometric Functions 7.31.2 Trigonometric Functions for Complex Arguments ---------------------------------------------------- -- Function: int gsl_sf_complex_sin_e (double ZR, double ZI, gsl_sf_result * SZR, gsl_sf_result * SZI) This function computes the complex sine, \sin(z_r + i z_i) storing the real and imaginary parts in SZR, SZI. -- Function: int gsl_sf_complex_cos_e (double ZR, double ZI, gsl_sf_result * CZR, gsl_sf_result * CZI) This function computes the complex cosine, \cos(z_r + i z_i) storing the real and imaginary parts in SZR, SZI. -- Function: int gsl_sf_complex_logsin_e (double ZR, double ZI, gsl_sf_result * LSZR, gsl_sf_result * LSZI) This function computes the logarithm of the complex sine, \log(\sin(z_r + i z_i)) storing the real and imaginary parts in SZR, SZI.  File: gsl-ref.info, Node: Hyperbolic Trigonometric Functions, Next: Conversion Functions, Prev: Trigonometric Functions for Complex Arguments, Up: Trigonometric Functions 7.31.3 Hyperbolic Trigonometric Functions ----------------------------------------- -- Function: double gsl_sf_lnsinh (double X) -- Function: int gsl_sf_lnsinh_e (double X, gsl_sf_result * RESULT) These routines compute \log(\sinh(x)) for x > 0. -- Function: double gsl_sf_lncosh (double X) -- Function: int gsl_sf_lncosh_e (double X, gsl_sf_result * RESULT) These routines compute \log(\cosh(x)) for any X.  File: gsl-ref.info, Node: Conversion Functions, Next: Restriction Functions, Prev: Hyperbolic Trigonometric Functions, Up: Trigonometric Functions 7.31.4 Conversion Functions --------------------------- -- Function: int gsl_sf_polar_to_rect (double R, double THETA, gsl_sf_result * X, gsl_sf_result * Y); This function converts the polar coordinates (R,THETA) to rectilinear coordinates (X,Y), x = r\cos(\theta), y = r\sin(\theta). -- Function: int gsl_sf_rect_to_polar (double X, double Y, gsl_sf_result * R, gsl_sf_result * THETA) This function converts the rectilinear coordinates (X,Y) to polar coordinates (R,THETA), such that x = r\cos(\theta), y = r\sin(\theta). The argument THETA lies in the range [-\pi, \pi].  File: gsl-ref.info, Node: Restriction Functions, Next: Trigonometric Functions With Error Estimates, Prev: Conversion Functions, Up: Trigonometric Functions 7.31.5 Restriction Functions ---------------------------- -- Function: double gsl_sf_angle_restrict_symm (double THETA) -- Function: int gsl_sf_angle_restrict_symm_e (double * THETA) These routines force the angle THETA to lie in the range (-\pi,\pi]. Note that the mathematical value of \pi is slightly greater than `M_PI', so the machine numbers `M_PI' and `-M_PI' are included in the range. -- Function: double gsl_sf_angle_restrict_pos (double THETA) -- Function: int gsl_sf_angle_restrict_pos_e (double * THETA) These routines force the angle THETA to lie in the range [0, 2\pi). Note that the mathematical value of 2\pi is slightly greater than `2*M_PI', so the machine number `2*M_PI' is included in the range.  File: gsl-ref.info, Node: Trigonometric Functions With Error Estimates, Prev: Restriction Functions, Up: Trigonometric Functions 7.31.6 Trigonometric Functions With Error Estimates --------------------------------------------------- -- Function: int gsl_sf_sin_err_e (double X, double DX, gsl_sf_result * RESULT) This routine computes the sine of an angle X with an associated absolute error DX, \sin(x \pm dx). Note that this function is provided in the error-handling form only since its purpose is to compute the propagated error. -- Function: int gsl_sf_cos_err_e (double X, double DX, gsl_sf_result * RESULT) This routine computes the cosine of an angle X with an associated absolute error DX, \cos(x \pm dx). Note that this function is provided in the error-handling form only since its purpose is to compute the propagated error.  File: gsl-ref.info, Node: Zeta Functions, Next: Special Functions Examples, Prev: Trigonometric Functions, Up: Special Functions 7.32 Zeta Functions =================== The Riemann zeta function is defined in Abramowitz & Stegun, Section 23.2. The functions described in this section are declared in the header file `gsl_sf_zeta.h'. * Menu: * Riemann Zeta Function:: * Riemann Zeta Function Minus One:: * Hurwitz Zeta Function:: * Eta Function::  File: gsl-ref.info, Node: Riemann Zeta Function, Next: Riemann Zeta Function Minus One, Up: Zeta Functions 7.32.1 Riemann Zeta Function ---------------------------- The Riemann zeta function is defined by the infinite sum \zeta(s) = \sum_{k=1}^\infty k^{-s}. -- Function: double gsl_sf_zeta_int (int N) -- Function: int gsl_sf_zeta_int_e (int N, gsl_sf_result * RESULT) These routines compute the Riemann zeta function \zeta(n) for integer N, n \ne 1. -- Function: double gsl_sf_zeta (double S) -- Function: int gsl_sf_zeta_e (double S, gsl_sf_result * RESULT) These routines compute the Riemann zeta function \zeta(s) for arbitrary S, s \ne 1.  File: gsl-ref.info, Node: Riemann Zeta Function Minus One, Next: Hurwitz Zeta Function, Prev: Riemann Zeta Function, Up: Zeta Functions 7.32.2 Riemann Zeta Function Minus One -------------------------------------- For large positive argument, the Riemann zeta function approaches one. In this region the fractional part is interesting, and therefore we need a function to evaluate it explicitly. -- Function: double gsl_sf_zetam1_int (int N) -- Function: int gsl_sf_zetam1_int_e (int N, gsl_sf_result * RESULT) These routines compute \zeta(n) - 1 for integer N, n \ne 1. -- Function: double gsl_sf_zetam1 (double S) -- Function: int gsl_sf_zetam1_e (double S, gsl_sf_result * RESULT) These routines compute \zeta(s) - 1 for arbitrary S, s \ne 1.  File: gsl-ref.info, Node: Hurwitz Zeta Function, Next: Eta Function, Prev: Riemann Zeta Function Minus One, Up: Zeta Functions 7.32.3 Hurwitz Zeta Function ---------------------------- The Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty (k+q)^{-s}. -- Function: double gsl_sf_hzeta (double S, double Q) -- Function: int gsl_sf_hzeta_e (double S, double Q, gsl_sf_result * RESULT) These routines compute the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.  File: gsl-ref.info, Node: Eta Function, Prev: Hurwitz Zeta Function, Up: Zeta Functions 7.32.4 Eta Function ------------------- The eta function is defined by \eta(s) = (1-2^{1-s}) \zeta(s). -- Function: double gsl_sf_eta_int (int N) -- Function: int gsl_sf_eta_int_e (int N, gsl_sf_result * RESULT) These routines compute the eta function \eta(n) for integer N. -- Function: double gsl_sf_eta (double S) -- Function: int gsl_sf_eta_e (double S, gsl_sf_result * RESULT) These routines compute the eta function \eta(s) for arbitrary S.  File: gsl-ref.info, Node: Special Functions Examples, Next: Special Functions References and Further Reading, Prev: Zeta Functions, Up: Special Functions 7.33 Examples ============= The following example demonstrates the use of the error handling form of the special functions, in this case to compute the Bessel function J_0(5.0), #include #include #include int main (void) { double x = 5.0; gsl_sf_result result; double expected = -0.17759677131433830434739701; int status = gsl_sf_bessel_J0_e (x, &result); printf ("status = %s\n", gsl_strerror(status)); printf ("J0(5.0) = %.18f\n" " +/- % .18f\n", result.val, result.err); printf ("exact = %.18f\n", expected); return status; } Here are the results of running the program, $ ./a.out status = success J0(5.0) = -0.177596771314338292 +/- 0.000000000000000193 exact = -0.177596771314338292 The next program computes the same quantity using the natural form of the function. In this case the error term RESULT.ERR and return status are not accessible. #include #include int main (void) { double x = 5.0; double expected = -0.17759677131433830434739701; double y = gsl_sf_bessel_J0 (x); printf ("J0(5.0) = %.18f\n", y); printf ("exact = %.18f\n", expected); return 0; } The results of the function are the same, $ ./a.out J0(5.0) = -0.177596771314338292 exact = -0.177596771314338292  File: gsl-ref.info, Node: Special Functions References and Further Reading, Prev: Special Functions Examples, Up: Special Functions 7.34 References and Further Reading =================================== The library follows the conventions of `Abramowitz & Stegun' where possible, Abramowitz & Stegun (eds.), `Handbook of Mathematical Functions' The following papers contain information on the algorithms used to compute the special functions, MISCFUN: A software package to compute uncommon special functions. `ACM Trans. Math. Soft.', vol. 22, 1996, 288-301 G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edition (Cambridge University Press, 1944). G. Nemeth, Mathematical Approximations of Special Functions, Nova Science Publishers, ISBN 1-56072-052-2 B.C. Carlson, Special Functions of Applied Mathematics (1977) W.J. Thompson, Atlas for Computing Mathematical Functions, John Wiley & Sons, New York (1997). Y.Y. Luke, Algorithms for the Computation of Mathematical Functions, Academic Press, New York (1977).  File: gsl-ref.info, Node: Vectors and Matrices, Next: Permutations, Prev: Special Functions, Up: Top 8 Vectors and Matrices ********************** The functions described in this chapter provide a simple vector and matrix interface to ordinary C arrays. The memory management of these arrays is implemented using a single underlying type, known as a block. By writing your functions in terms of vectors and matrices you can pass a single structure containing both data and dimensions as an argument without needing additional function parameters. The structures are compatible with the vector and matrix formats used by BLAS routines. * Menu: * Data types:: * Blocks:: * Vectors:: * Matrices:: * Vector and Matrix References and Further Reading::  File: gsl-ref.info, Node: Data types, Next: Blocks, Up: Vectors and Matrices 8.1 Data types ============== All the functions are available for each of the standard data-types. The versions for `double' have the prefix `gsl_block', `gsl_vector' and `gsl_matrix'. Similarly the versions for single-precision `float' arrays have the prefix `gsl_block_float', `gsl_vector_float' and `gsl_matrix_float'. The full list of available types is given below, gsl_block double gsl_block_float float gsl_block_long_double long double gsl_block_int int gsl_block_uint unsigned int gsl_block_long long gsl_block_ulong unsigned long gsl_block_short short gsl_block_ushort unsigned short gsl_block_char char gsl_block_uchar unsigned char gsl_block_complex complex double gsl_block_complex_float complex float gsl_block_complex_long_double complex long double Corresponding types exist for the `gsl_vector' and `gsl_matrix' functions.  File: gsl-ref.info, Node: Blocks, Next: Vectors, Prev: Data types, Up: Vectors and Matrices 8.2 Blocks ========== For consistency all memory is allocated through a `gsl_block' structure. The structure contains two components, the size of an area of memory and a pointer to the memory. The `gsl_block' structure looks like this, typedef struct { size_t size; double * data; } gsl_block; Vectors and matrices are made by "slicing" an underlying block. A slice is a set of elements formed from an initial offset and a combination of indices and step-sizes. In the case of a matrix the step-size for the column index represents the row-length. The step-size for a vector is known as the "stride". The functions for allocating and deallocating blocks are defined in `gsl_block.h' * Menu: * Block allocation:: * Reading and writing blocks:: * Example programs for blocks::  File: gsl-ref.info, Node: Block allocation, Next: Reading and writing blocks, Up: Blocks 8.2.1 Block allocation ---------------------- The functions for allocating memory to a block follow the style of `malloc' and `free'. In addition they also perform their own error checking. If there is insufficient memory available to allocate a block then the functions call the GSL error handler (with an error number of `GSL_ENOMEM') in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every `alloc'. -- Function: gsl_block * gsl_block_alloc (size_t N) This function allocates memory for a block of N double-precision elements, returning a pointer to the block struct. The block is not initialized and so the values of its elements are undefined. Use the function `gsl_block_calloc' if you want to ensure that all the elements are initialized to zero. A null pointer is returned if insufficient memory is available to create the block. -- Function: gsl_block * gsl_block_calloc (size_t N) This function allocates memory for a block and initializes all the elements of the block to zero. -- Function: void gsl_block_free (gsl_block * B) This function frees the memory used by a block B previously allocated with `gsl_block_alloc' or `gsl_block_calloc'. The block B must be a valid block object (a null pointer is not allowed).  File: gsl-ref.info, Node: Reading and writing blocks, Next: Example programs for blocks, Prev: Block allocation, Up: Blocks 8.2.2 Reading and writing blocks -------------------------------- The library provides functions for reading and writing blocks to a file as binary data or formatted text. -- Function: int gsl_block_fwrite (FILE * STREAM, const gsl_block * B) This function writes the elements of the block B to the stream STREAM in binary format. The return value is 0 for success and `GSL_EFAILED' if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures. -- Function: int gsl_block_fread (FILE * STREAM, gsl_block * B) This function reads into the block B from the open stream STREAM in binary format. The block B must be preallocated with the correct length since the function uses the size of B to determine how many bytes to read. The return value is 0 for success and `GSL_EFAILED' if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture. -- Function: int gsl_block_fprintf (FILE * STREAM, const gsl_block * B, const char * FORMAT) This function writes the elements of the block B line-by-line to the stream STREAM using the format specifier FORMAT, which should be one of the `%g', `%e' or `%f' formats for floating point numbers and `%d' for integers. The function returns 0 for success and `GSL_EFAILED' if there was a problem writing to the file. -- Function: int gsl_block_fscanf (FILE * STREAM, gsl_block * B) This function reads formatted data from the stream STREAM into the block B. The block B must be preallocated with the correct length since the function uses the size of B to determine how many numbers to read. The function returns 0 for success and `GSL_EFAILED' if there was a problem reading from the file.  File: gsl-ref.info, Node: Example programs for blocks, Prev: Reading and writing blocks, Up: Blocks 8.2.3 Example programs for blocks --------------------------------- The following program shows how to allocate a block, #include #include int main (void) { gsl_block * b = gsl_block_alloc (100); printf ("length of block = %u\n", b->size); printf ("block data address = %#x\n", b->data); gsl_block_free (b); return 0; } Here is the output from the program, length of block = 100 block data address = 0x804b0d8  File: gsl-ref.info, Node: Vectors, Next: Matrices, Prev: Blocks, Up: Vectors and Matrices 8.3 Vectors =========== Vectors are defined by a `gsl_vector' structure which describes a slice of a block. Different vectors can be created which point to the same block. A vector slice is a set of equally-spaced elements of an area of memory. The `gsl_vector' structure contains five components, the "size", the "stride", a pointer to the memory where the elements are stored, DATA, a pointer to the block owned by the vector, BLOCK, if any, and an ownership flag, OWNER. The structure is very simple and looks like this, typedef struct { size_t size; size_t stride; double * data; gsl_block * block; int owner; } gsl_vector; The SIZE is simply the number of vector elements. The range of valid indices runs from 0 to `size-1'. The STRIDE is the step-size from one element to the next in physical memory, measured in units of the appropriate datatype. The pointer DATA gives the location of the first element of the vector in memory. The pointer BLOCK stores the location of the memory block in which the vector elements are located (if any). If the vector owns this block then the OWNER field is set to one and the block will be deallocated when the vector is freed. If the vector points to a block owned by another object then the OWNER field is zero and any underlying block will not be deallocated with the vector. The functions for allocating and accessing vectors are defined in `gsl_vector.h' * Menu: * Vector allocation:: * Accessing vector elements:: * Initializing vector elements:: * Reading and writing vectors:: * Vector views:: * Copying vectors:: * Exchanging elements:: * Vector operations:: * Finding maximum and minimum elements of vectors:: * Vector properties:: * Example programs for vectors::  File: gsl-ref.info, Node: Vector allocation, Next: Accessing vector elements, Up: Vectors 8.3.1 Vector allocation ----------------------- The functions for allocating memory to a vector follow the style of `malloc' and `free'. In addition they also perform their own error checking. If there is insufficient memory available to allocate a vector then the functions call the GSL error handler (with an error number of `GSL_ENOMEM') in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every `alloc'. -- Function: gsl_vector * gsl_vector_alloc (size_t N) This function creates a vector of length N, returning a pointer to a newly initialized vector struct. A new block is allocated for the elements of the vector, and stored in the BLOCK component of the vector struct. The block is "owned" by the vector, and will be deallocated when the vector is deallocated. -- Function: gsl_vector * gsl_vector_calloc (size_t N) This function allocates memory for a vector of length N and initializes all the elements of the vector to zero. -- Function: void gsl_vector_free (gsl_vector * V) This function frees a previously allocated vector V. If the vector was created using `gsl_vector_alloc' then the block underlying the vector will also be deallocated. If the vector has been created from another object then the memory is still owned by that object and will not be deallocated. The vector V must be a valid vector object (a null pointer is not allowed).  File: gsl-ref.info, Node: Accessing vector elements, Next: Initializing vector elements, Prev: Vector allocation, Up: Vectors 8.3.2 Accessing vector elements ------------------------------- Unlike FORTRAN compilers, C compilers do not usually provide support for range checking of vectors and matrices. Range checking is available in the GNU C Compiler bounds-checking extension, but it is not part of the default installation of GCC. The functions `gsl_vector_get' and `gsl_vector_set' can perform portable range checking for you and report an error if you attempt to access elements outside the allowed range. The functions for accessing the elements of a vector or matrix are defined in `gsl_vector.h' and declared `extern inline' to eliminate function-call overhead. You must compile your program with the macro `HAVE_INLINE' defined to use these functions. If necessary you can turn off range checking completely without modifying any source files by recompiling your program with the preprocessor definition `GSL_RANGE_CHECK_OFF'. Provided your compiler supports inline functions the effect of turning off range checking is to replace calls to `gsl_vector_get(v,i)' by `v->data[i*v->stride]' and calls to `gsl_vector_set(v,i,x)' by `v->data[i*v->stride]=x'. Thus there should be no performance penalty for using the range checking functions when range checking is turned off. -- Function: double gsl_vector_get (const gsl_vector * V, size_t I) This function returns the I-th element of a vector V. If I lies outside the allowed range of 0 to N-1 then the error handler is invoked and 0 is returned. -- Function: void gsl_vector_set (gsl_vector * V, size_t I, double X) This function sets the value of the I-th element of a vector V to X. If I lies outside the allowed range of 0 to N-1 then the error handler is invoked. -- Function: double * gsl_vector_ptr (gsl_vector * V, size_t I) -- Function: const double * gsl_vector_const_ptr (const gsl_vector * V, size_t I) These functions return a pointer to the I-th element of a vector V. If I lies outside the allowed range of 0 to N-1 then the error handler is invoked and a null pointer is returned.  File: gsl-ref.info, Node: Initializing vector elements, Next: Reading and writing vectors, Prev: Accessing vector elements, Up: Vectors 8.3.3 Initializing vector elements ---------------------------------- -- Function: void gsl_vector_set_all (gsl_vector * V, double X) This function sets all the elements of the vector V to the value X. -- Function: void gsl_vector_set_zero (gsl_vector * V) This function sets all the elements of the vector V to zero. -- Function: int gsl_vector_set_basis (gsl_vector * V, size_t I) This function makes a basis vector by setting all the elements of the vector V to zero except for the I-th element which is set to one.  File: gsl-ref.info, Node: Reading and writing vectors, Next: Vector views, Prev: Initializing vector elements, Up: Vectors 8.3.4 Reading and writing vectors --------------------------------- The library provides functions for reading and writing vectors to a file as binary data or formatted text. -- Function: int gsl_vector_fwrite (FILE * STREAM, const gsl_vector * V) This function writes the elements of the vector V to the stream STREAM in binary format. The return value is 0 for success and `GSL_EFAILED' if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures. -- Function: int gsl_vector_fread (FILE * STREAM, gsl_vector * V) This function reads into the vector V from the open stream STREAM in binary format. The vector V must be preallocated with the correct length since the function uses the size of V to determine how many bytes to read. The return value is 0 for success and `GSL_EFAILED' if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture. -- Function: int gsl_vector_fprintf (FILE * STREAM, const gsl_vector * V, const char * FORMAT) This function writes the elements of the vector V line-by-line to the stream STREAM using the format specifier FORMAT, which should be one of the `%g', `%e' or `%f' formats for floating point numbers and `%d' for integers. The function returns 0 for success and `GSL_EFAILED' if there was a problem writing to the file. -- Function: int gsl_vector_fscanf (FILE * STREAM, gsl_vector * V) This function reads formatted data from the stream STREAM into the vector V. The vector V must be preallocated with the correct length since the function uses the size of V to determine how many numbers to read. The function returns 0 for success and `GSL_EFAILED' if there was a problem reading from the file.  File: gsl-ref.info, Node: Vector views, Next: Copying vectors, Prev: Reading and writing vectors, Up: Vectors 8.3.5 Vector views ------------------ In addition to creating vectors from slices of blocks it is also possible to slice vectors and create vector views. For example, a subvector of another vector can be described with a view, or two views can be made which provide access to the even and odd elements of a vector. A vector view is a temporary object, stored on the stack, which can be used to operate on a subset of vector elements. Vector views can be defined for both constant and non-constant vectors, using separate types that preserve constness. A vector view has the type `gsl_vector_view' and a constant vector view has the type `gsl_vector_const_view'. In both cases the elements of the view can be accessed as a `gsl_vector' using the `vector' component of the view object. A pointer to a vector of type `gsl_vector *' or `const gsl_vector *' can be obtained by taking the address of this component with the `&' operator. When using this pointer it is important to ensure that the view itself remains in scope--the simplest way to do so is by always writing the pointer as `&'VIEW`.vector', and never storing this value in another variable. -- Function: gsl_vector_view gsl_vector_subvector (gsl_vector * V, size_t OFFSET, size_t N) -- Function: gsl_vector_const_view gsl_vector_const_subvector (const gsl_vector * V, size_t OFFSET, size_t N) These functions return a vector view of a subvector of another vector V. The start of the new vector is offset by OFFSET elements from the start of the original vector. The new vector has N elements. Mathematically, the I-th element of the new vector V' is given by, v'(i) = v->data[(offset + i)*v->stride] where the index I runs from 0 to `n-1'. The `data' pointer of the returned vector struct is set to null if the combined parameters (OFFSET,N) overrun the end of the original vector. The new vector is only a view of the block underlying the original vector, V. The block containing the elements of V is not owned by the new vector. When the view goes out of scope the original vector V and its block will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use. The function `gsl_vector_const_subvector' is equivalent to `gsl_vector_subvector' but can be used for vectors which are declared `const'. -- Function: gsl_vector_view gsl_vector_subvector_with_stride (gsl_vector * V, size_t OFFSET, size_t STRIDE, size_t N) -- Function: gsl_vector_const_view gsl_vector_const_subvector_with_stride (const gsl_vector * V, size_t OFFSET, size_t STRIDE, size_t N) These functions return a vector view of a subvector of another vector V with an additional stride argument. The subvector is formed in the same way as for `gsl_vector_subvector' but the new vector has N elements with a step-size of STRIDE from one element to the next in the original vector. Mathematically, the I-th element of the new vector V' is given by, v'(i) = v->data[(offset + i*stride)*v->stride] where the index I runs from 0 to `n-1'. Note that subvector views give direct access to the underlying elements of the original vector. For example, the following code will zero the even elements of the vector `v' of length `n', while leaving the odd elements untouched, gsl_vector_view v_even = gsl_vector_subvector_with_stride (v, 0, 2, n/2); gsl_vector_set_zero (&v_even.vector); A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector would be, using `&'VIEW`.vector'. For example, the following code computes the norm of the odd elements of `v' using the BLAS routine DNRM2, gsl_vector_view v_odd = gsl_vector_subvector_with_stride (v, 1, 2, n/2); double r = gsl_blas_dnrm2 (&v_odd.vector); The function `gsl_vector_const_subvector_with_stride' is equivalent to `gsl_vector_subvector_with_stride' but can be used for vectors which are declared `const'. -- Function: gsl_vector_view gsl_vector_complex_real (gsl_vector_complex * V) -- Function: gsl_vector_const_view gsl_vector_complex_const_real (const gsl_vector_complex * V) These functions return a vector view of the real parts of the complex vector V. The function `gsl_vector_complex_const_real' is equivalent to `gsl_vector_complex_real' but can be used for vectors which are declared `const'. -- Function: gsl_vector_view gsl_vector_complex_imag (gsl_vector_complex * V) -- Function: gsl_vector_const_view gsl_vector_complex_const_imag (const gsl_vector_complex * V) These functions return a vector view of the imaginary parts of the complex vector V. The function `gsl_vector_complex_const_imag' is equivalent to `gsl_vector_complex_imag' but can be used for vectors which are declared `const'. -- Function: gsl_vector_view gsl_vector_view_array (double * BASE, size_t N) -- Function: gsl_vector_const_view gsl_vector_const_view_array (const double * BASE, size_t N) These functions return a vector view of an array. The start of the new vector is given by BASE and has N elements. Mathematically, the I-th element of the new vector V' is given by, v'(i) = base[i] where the index I runs from 0 to `n-1'. The array containing the elements of V is not owned by the new vector view. When the view goes out of scope the original array will continue to exist. The original memory can only be deallocated by freeing the original pointer BASE. Of course, the original array should not be deallocated while the view is still in use. The function `gsl_vector_const_view_array' is equivalent to `gsl_vector_view_array' but can be used for arrays which are declared `const'. -- Function: gsl_vector_view gsl_vector_view_array_with_stride (double * BASE, size_t STRIDE, size_t N) -- Function: gsl_vector_const_view gsl_vector_const_view_array_with_stride (const double * BASE, size_t STRIDE, size_t N) These functions return a vector view of an array BASE with an additional stride argument. The subvector is formed in the same way as for `gsl_vector_view_array' but the new vector has N elements with a step-size of STRIDE from one element to the next in the original array. Mathematically, the I-th element of the new vector V' is given by, v'(i) = base[i*stride] where the index I runs from 0 to `n-1'. Note that the view gives direct access to the underlying elements of the original array. A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector would be, using `&'VIEW`.vector'. The function `gsl_vector_const_view_array_with_stride' is equivalent to `gsl_vector_view_array_with_stride' but can be used for arrays which are declared `const'.  File: gsl-ref.info, Node: Copying vectors, Next: Exchanging elements, Prev: Vector views, Up: Vectors 8.3.6 Copying vectors --------------------- Common operations on vectors such as addition and multiplication are available in the BLAS part of the library (*note BLAS Support::). However, it is useful to have a small number of utility functions which do not require the full BLAS code. The following functions fall into this category. -- Function: int gsl_vector_memcpy (gsl_vector * DEST, const gsl_vector * SRC) This function copies the elements of the vector SRC into the vector DEST. The two vectors must have the same length. -- Function: int gsl_vector_swap (gsl_vector * V, gsl_vector * W) This function exchanges the elements of the vectors V and W by copying. The two vectors must have the same length.  File: gsl-ref.info, Node: Exchanging elements, Next: Vector operations, Prev: Copying vectors, Up: Vectors 8.3.7 Exchanging elements ------------------------- The following function can be used to exchange, or permute, the elements of a vector. -- Function: int gsl_vector_swap_elements (gsl_vector * V, size_t I, size_t J) This function exchanges the I-th and J-th elements of the vector V in-place. -- Function: int gsl_vector_reverse (gsl_vector * V) This function reverses the order of the elements of the vector V.  File: gsl-ref.info, Node: Vector operations, Next: Finding maximum and minimum elements of vectors, Prev: Exchanging elements, Up: Vectors 8.3.8 Vector operations ----------------------- The following operations are only defined for real vectors. -- Function: int gsl_vector_add (gsl_vector * A, const gsl_vector * B) This function adds the elements of vector B to the elements of vector A, a'_i = a_i + b_i. The two vectors must have the same length. -- Function: int gsl_vector_sub (gsl_vector * A, const gsl_vector * B) This function subtracts the elements of vector B from the elements of vector A, a'_i = a_i - b_i. The two vectors must have the same length. -- Function: int gsl_vector_mul (gsl_vector * A, const gsl_vector * B) This function multiplies the elements of vector A by the elements of vector B, a'_i = a_i * b_i. The two vectors must have the same length. -- Function: int gsl_vector_div (gsl_vector * A, const gsl_vector * B) This function divides the elements of vector A by the elements of vector B, a'_i = a_i / b_i. The two vectors must have the same length. -- Function: int gsl_vector_scale (gsl_vector * A, const double X) This function multiplies the elements of vector A by the constant factor X, a'_i = x a_i. -- Function: int gsl_vector_add_constant (gsl_vector * A, const double X) This function adds the constant value X to the elements of the vector A, a'_i = a_i + x.  File: gsl-ref.info, Node: Finding maximum and minimum elements of vectors, Next: Vector properties, Prev: Vector operations, Up: Vectors 8.3.9 Finding maximum and minimum elements of vectors ----------------------------------------------------- -- Function: double gsl_vector_max (const gsl_vector * V) This function returns the maximum value in the vector V. -- Function: double gsl_vector_min (const gsl_vector * V) This function returns the minimum value in the vector V. -- Function: void gsl_vector_minmax (const gsl_vector * V, double * MIN_OUT, double * MAX_OUT) This function returns the minimum and maximum values in the vector V, storing them in MIN_OUT and MAX_OUT. -- Function: size_t gsl_vector_max_index (const gsl_vector * V) This function returns the index of the maximum value in the vector V. When there are several equal maximum elements then the lowest index is returned. -- Function: size_t gsl_vector_min_index (const gsl_vector * V) This function returns the index of the minimum value in the vector V. When there are several equal minimum elements then the lowest index is returned. -- Function: void gsl_vector_minmax_index (const gsl_vector * V, size_t * IMIN, size_t * IMAX) This function returns the indices of the minimum and maximum values in the vector V, storing them in IMIN and IMAX. When there are several equal minimum or maximum elements then the lowest indices are returned.  File: gsl-ref.info, Node: Vector properties, Next: Example programs for vectors, Prev: Finding maximum and minimum elements of vectors, Up: Vectors 8.3.10 Vector properties ------------------------ -- Function: int gsl_vector_isnull (const gsl_vector * V) -- Function: int gsl_vector_ispos (const gsl_vector * V) -- Function: int gsl_vector_isneg (const gsl_vector * V) These functions return 1 if all the elements of the vector V are zero, strictly positive, or strictly negative respectively, and 0 otherwise. To test for a non-negative vector, use the expression `!gsl_vector_isneg(v)'.  File: gsl-ref.info, Node: Example programs for vectors, Prev: Vector properties, Up: Vectors 8.3.11 Example programs for vectors ----------------------------------- This program shows how to allocate, initialize and read from a vector using the functions `gsl_vector_alloc', `gsl_vector_set' and `gsl_vector_get'. #include #include 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 #include 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 #include int main (void) { int i; gsl_vector * v = gsl_vector_alloc (10); { FILE * f = fopen ("test.dat", "r"); gsl_vector_fscanf (f, v); fclose (f); } for (i = 0; i < 10; i++) { printf ("%g\n", gsl_vector_get(v, i)); } gsl_vector_free (v); return 0; }  File: gsl-ref.info, Node: Matrices, Next: Vector and Matrix References and Further Reading, Prev: Vectors, Up: Vectors and Matrices 8.4 Matrices ============ Matrices are defined by a `gsl_matrix' structure which describes a generalized slice of a block. Like a vector it represents a set of elements in an area of memory, but uses two indices instead of one. The `gsl_matrix' structure contains six components, the two dimensions of the matrix, a physical dimension, a pointer to the memory where the elements of the matrix are stored, DATA, a pointer to the block owned by the matrix BLOCK, if any, and an ownership flag, OWNER. The physical dimension determines the memory layout and can differ from the matrix dimension to allow the use of submatrices. The `gsl_matrix' structure is very simple and looks like this, typedef struct { size_t size1; size_t size2; size_t tda; double * data; gsl_block * block; int owner; } gsl_matrix; Matrices are stored in row-major order, meaning that each row of elements forms a contiguous block in memory. This is the standard "C-language ordering" of two-dimensional arrays. Note that FORTRAN stores arrays in column-major order. The number of rows is SIZE1. The range of valid row indices runs from 0 to `size1-1'. Similarly SIZE2 is the number of columns. The range of valid column indices runs from 0 to `size2-1'. The physical row dimension TDA, or "trailing dimension", specifies the size of a row of the matrix as laid out in memory. For example, in the following matrix SIZE1 is 3, SIZE2 is 4, and TDA is 8. The physical memory layout of the matrix begins in the top left hand-corner and proceeds from left to right along each row in turn. 00 01 02 03 XX XX XX XX 10 11 12 13 XX XX XX XX 20 21 22 23 XX XX XX XX Each unused memory location is represented by "`XX'". The pointer DATA gives the location of the first element of the matrix in memory. The pointer BLOCK stores the location of the memory block in which the elements of the matrix are located (if any). If the matrix owns this block then the OWNER field is set to one and the block will be deallocated when the matrix is freed. If the matrix is only a slice of a block owned by another object then the OWNER field is zero and any underlying block will not be freed. The functions for allocating and accessing matrices are defined in `gsl_matrix.h' * Menu: * Matrix allocation:: * Accessing matrix elements:: * Initializing matrix elements:: * Reading and writing matrices:: * Matrix views:: * Creating row and column views:: * Copying matrices:: * Copying rows and columns:: * Exchanging rows and columns:: * Matrix operations:: * Finding maximum and minimum elements of matrices:: * Matrix properties:: * Example programs for matrices::  File: gsl-ref.info, Node: Matrix allocation, Next: Accessing matrix elements, Up: Matrices 8.4.1 Matrix allocation ----------------------- The functions for allocating memory to a matrix follow the style of `malloc' and `free'. They also perform their own error checking. If there is insufficient memory available to allocate a vector then the functions call the GSL error handler (with an error number of `GSL_ENOMEM') in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every `alloc'. -- Function: gsl_matrix * gsl_matrix_alloc (size_t N1, size_t N2) This function creates a matrix of size N1 rows by N2 columns, returning a pointer to a newly initialized matrix struct. A new block is allocated for the elements of the matrix, and stored in the BLOCK component of the matrix struct. The block is "owned" by the matrix, and will be deallocated when the matrix is deallocated. -- Function: gsl_matrix * gsl_matrix_calloc (size_t N1, size_t N2) This function allocates memory for a matrix of size N1 rows by N2 columns and initializes all the elements of the matrix to zero. -- Function: void gsl_matrix_free (gsl_matrix * M) This function frees a previously allocated matrix M. If the matrix was created using `gsl_matrix_alloc' then the block underlying the matrix will also be deallocated. If the matrix has been created from another object then the memory is still owned by that object and will not be deallocated. The matrix M must be a valid matrix object (a null pointer is not allowed).  File: gsl-ref.info, Node: Accessing matrix elements, Next: Initializing matrix elements, Prev: Matrix allocation, Up: Matrices 8.4.2 Accessing matrix elements ------------------------------- The functions for accessing the elements of a matrix use the same range checking system as vectors. You can turn off range checking by recompiling your program with the preprocessor definition `GSL_RANGE_CHECK_OFF'. The elements of the matrix are stored in "C-order", where the second index moves continuously through memory. More precisely, the element accessed by the function `gsl_matrix_get(m,i,j)' and `gsl_matrix_set(m,i,j,x)' is m->data[i * m->tda + j] where TDA is the physical row-length of the matrix. -- Function: double gsl_matrix_get (const gsl_matrix * M, size_t I, size_t J) This function returns the (i,j)-th element of a matrix M. If I or J lie outside the allowed range of 0 to N1-1 and 0 to N2-1 then the error handler is invoked and 0 is returned. -- Function: void gsl_matrix_set (gsl_matrix * M, size_t I, size_t J, double X) This function sets the value of the (i,j)-th element of a matrix M to X. If I or J lies outside the allowed range of 0 to N1-1 and 0 to N2-1 then the error handler is invoked. -- Function: double * gsl_matrix_ptr (gsl_matrix * M, size_t I, size_t J) -- Function: const double * gsl_matrix_const_ptr (const gsl_matrix * M, size_t I, size_t J) These functions return a pointer to the (i,j)-th element of a matrix M. If I or J lie outside the allowed range of 0 to N1-1 and 0 to N2-1 then the error handler is invoked and a null pointer is returned.  File: gsl-ref.info, Node: Initializing matrix elements, Next: Reading and writing matrices, Prev: Accessing matrix elements, Up: Matrices 8.4.3 Initializing matrix elements ---------------------------------- -- Function: void gsl_matrix_set_all (gsl_matrix * M, double X) This function sets all the elements of the matrix M to the value X. -- Function: void gsl_matrix_set_zero (gsl_matrix * M) This function sets all the elements of the matrix M to zero. -- Function: void gsl_matrix_set_identity (gsl_matrix * M) This function sets the elements of the matrix M to the corresponding elements of the identity matrix, m(i,j) = \delta(i,j), i.e. a unit diagonal with all off-diagonal elements zero. This applies to both square and rectangular matrices.  File: gsl-ref.info, Node: Reading and writing matrices, Next: Matrix views, Prev: Initializing matrix elements, Up: Matrices 8.4.4 Reading and writing matrices ---------------------------------- The library provides functions for reading and writing matrices to a file as binary data or formatted text. -- Function: int gsl_matrix_fwrite (FILE * STREAM, const gsl_matrix * M) This function writes the elements of the matrix M to the stream STREAM in binary format. The return value is 0 for success and `GSL_EFAILED' if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures. -- Function: int gsl_matrix_fread (FILE * STREAM, gsl_matrix * M) This function reads into the matrix M from the open stream STREAM in binary format. The matrix M must be preallocated with the correct dimensions since the function uses the size of M to determine how many bytes to read. The return value is 0 for success and `GSL_EFAILED' if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture. -- Function: int gsl_matrix_fprintf (FILE * STREAM, const gsl_matrix * M, const char * FORMAT) This function writes the elements of the matrix M line-by-line to the stream STREAM using the format specifier FORMAT, which should be one of the `%g', `%e' or `%f' formats for floating point numbers and `%d' for integers. The function returns 0 for success and `GSL_EFAILED' if there was a problem writing to the file. -- Function: int gsl_matrix_fscanf (FILE * STREAM, gsl_matrix * M) This function reads formatted data from the stream STREAM into the matrix M. The matrix M must be preallocated with the correct dimensions since the function uses the size of M to determine how many numbers to read. The function returns 0 for success and `GSL_EFAILED' if there was a problem reading from the file.  File: gsl-ref.info, Node: Matrix views, Next: Creating row and column views, Prev: Reading and writing matrices, Up: Matrices 8.4.5 Matrix views ------------------ A matrix view is a temporary object, stored on the stack, which can be used to operate on a subset of matrix elements. Matrix views can be defined for both constant and non-constant matrices using separate types that preserve constness. A matrix view has the type `gsl_matrix_view' and a constant matrix view has the type `gsl_matrix_const_view'. In both cases the elements of the view can by accessed using the `matrix' component of the view object. A pointer `gsl_matrix *' or `const gsl_matrix *' can be obtained by taking the address of the `matrix' component with the `&' operator. In addition to matrix views it is also possible to create vector views of a matrix, such as row or column views. -- Function: gsl_matrix_view gsl_matrix_submatrix (gsl_matrix * M, size_t K1, size_t K2, size_t N1, size_t N2) -- Function: gsl_matrix_const_view gsl_matrix_const_submatrix (const gsl_matrix * M, size_t K1, size_t K2, size_t N1, size_t N2) These functions return a matrix view of a submatrix of the matrix M. The upper-left element of the submatrix is the element (K1,K2) of the original matrix. The submatrix has N1 rows and N2 columns. The physical number of columns in memory given by TDA is unchanged. Mathematically, the (i,j)-th element of the new matrix is given by, m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j] where the index I runs from 0 to `n1-1' and the index J runs from 0 to `n2-1'. The `data' pointer of the returned matrix struct is set to null if the combined parameters (I,J,N1,N2,TDA) overrun the ends of the original matrix. The new matrix view is only a view of the block underlying the existing matrix, M. The block containing the elements of M is not owned by the new matrix view. When the view goes out of scope the original matrix M and its block will continue to exist. The original memory can only be deallocated by freeing the original matrix. Of course, the original matrix should not be deallocated while the view is still in use. The function `gsl_matrix_const_submatrix' is equivalent to `gsl_matrix_submatrix' but can be used for matrices which are declared `const'. -- Function: gsl_matrix_view gsl_matrix_view_array (double * BASE, size_t N1, size_t N2) -- Function: gsl_matrix_const_view gsl_matrix_const_view_array (const double * BASE, size_t N1, size_t N2) These functions return a matrix view of the array BASE. The matrix has N1 rows and N2 columns. The physical number of columns in memory is also given by N2. Mathematically, the (i,j)-th element of the new matrix is given by, m'(i,j) = base[i*n2 + j] where the index I runs from 0 to `n1-1' and the index J runs from 0 to `n2-1'. The new matrix is only a view of the array BASE. When the view goes out of scope the original array BASE will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use. The function `gsl_matrix_const_view_array' is equivalent to `gsl_matrix_view_array' but can be used for matrices which are declared `const'. -- Function: gsl_matrix_view gsl_matrix_view_array_with_tda (double * BASE, size_t N1, size_t N2, size_t TDA) -- Function: gsl_matrix_const_view gsl_matrix_const_view_array_with_tda (const double * BASE, size_t N1, size_t N2, size_t TDA) These functions return a matrix view of the array BASE with a physical number of columns TDA which may differ from the corresponding dimension of the matrix. The matrix has N1 rows and N2 columns, and the physical number of columns in memory is given by TDA. Mathematically, the (i,j)-th element of the new matrix is given by, m'(i,j) = base[i*tda + j] where the index I runs from 0 to `n1-1' and the index J runs from 0 to `n2-1'. The new matrix is only a view of the array BASE. When the view goes out of scope the original array BASE will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use. The function `gsl_matrix_const_view_array_with_tda' is equivalent to `gsl_matrix_view_array_with_tda' but can be used for matrices which are declared `const'. -- Function: gsl_matrix_view gsl_matrix_view_vector (gsl_vector * V, size_t N1, size_t N2) -- Function: gsl_matrix_const_view gsl_matrix_const_view_vector (const gsl_vector * V, size_t N1, size_t N2) These functions return a matrix view of the vector V. The matrix has N1 rows and N2 columns. The vector must have unit stride. The physical number of columns in memory is also given by N2. Mathematically, the (i,j)-th element of the new matrix is given by, m'(i,j) = v->data[i*n2 + j] where the index I runs from 0 to `n1-1' and the index J runs from 0 to `n2-1'. The new matrix is only a view of the vector V. When the view goes out of scope the original vector V will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use. The function `gsl_matrix_const_view_vector' is equivalent to `gsl_matrix_view_vector' but can be used for matrices which are declared `const'. -- Function: gsl_matrix_view gsl_matrix_view_vector_with_tda (gsl_vector * V, size_t N1, size_t N2, size_t TDA) -- Function: gsl_matrix_const_view gsl_matrix_const_view_vector_with_tda (const gsl_vector * V, size_t N1, size_t N2, size_t TDA) These functions return a matrix view of the vector V with a physical number of columns TDA which may differ from the corresponding matrix dimension. The vector must have unit stride. The matrix has N1 rows and N2 columns, and the physical number of columns in memory is given by TDA. Mathematically, the (i,j)-th element of the new matrix is given by, m'(i,j) = v->data[i*tda + j] where the index I runs from 0 to `n1-1' and the index J runs from 0 to `n2-1'. The new matrix is only a view of the vector V. When the view goes out of scope the original vector V will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use. The function `gsl_matrix_const_view_vector_with_tda' is equivalent to `gsl_matrix_view_vector_with_tda' but can be used for matrices which are declared `const'.  File: gsl-ref.info, Node: Creating row and column views, Next: Copying matrices, Prev: Matrix views, Up: Matrices 8.4.6 Creating row and column views ----------------------------------- In general there are two ways to access an object, by reference or by copying. The functions described in this section create vector views which allow access to a row or column of a matrix by reference. Modifying elements of the view is equivalent to modifying the matrix, since both the vector view and the matrix point to the same memory block. -- Function: gsl_vector_view gsl_matrix_row (gsl_matrix * M, size_t I) -- Function: gsl_vector_const_view gsl_matrix_const_row (const gsl_matrix * M, size_t I) These functions return a vector view of the I-th row of the matrix M. The `data' pointer of the new vector is set to null if I is out of range. The function `gsl_vector_const_row' is equivalent to `gsl_matrix_row' but can be used for matrices which are declared `const'. -- Function: gsl_vector_view gsl_matrix_column (gsl_matrix * M, size_t J) -- Function: gsl_vector_const_view gsl_matrix_const_column (const gsl_matrix * M, size_t J) These functions return a vector view of the J-th column of the matrix M. The `data' pointer of the new vector is set to null if J is out of range. The function `gsl_vector_const_column' is equivalent to `gsl_matrix_column' but can be used for matrices which are declared `const'. -- Function: gsl_vector_view gsl_matrix_diagonal (gsl_matrix * M) -- Function: gsl_vector_const_view gsl_matrix_const_diagonal (const gsl_matrix * M) These functions returns a vector view of the diagonal of the matrix M. The matrix M is not required to be square. For a rectangular matrix the length of the diagonal is the same as the smaller dimension of the matrix. The function `gsl_matrix_const_diagonal' is equivalent to `gsl_matrix_diagonal' but can be used for matrices which are declared `const'. -- Function: gsl_vector_view gsl_matrix_subdiagonal (gsl_matrix * M, size_t K) -- Function: gsl_vector_const_view gsl_matrix_const_subdiagonal (const gsl_matrix * M, size_t K) These functions return a vector view of the K-th subdiagonal of the matrix M. The matrix M is not required to be square. The diagonal of the matrix corresponds to k = 0. The function `gsl_matrix_const_subdiagonal' is equivalent to `gsl_matrix_subdiagonal' but can be used for matrices which are declared `const'. -- Function: gsl_vector_view gsl_matrix_superdiagonal (gsl_matrix * M, size_t K) -- Function: gsl_vector_const_view gsl_matrix_const_superdiagonal (const gsl_matrix * M, size_t K) These functions return a vector view of the K-th superdiagonal of the matrix M. The matrix M is not required to be square. The diagonal of the matrix corresponds to k = 0. The function `gsl_matrix_const_superdiagonal' is equivalent to `gsl_matrix_superdiagonal' but can be used for matrices which are declared `const'.  File: gsl-ref.info, Node: Copying matrices, Next: Copying rows and columns, Prev: Creating row and column views, Up: Matrices 8.4.7 Copying matrices ---------------------- -- Function: int gsl_matrix_memcpy (gsl_matrix * DEST, const gsl_matrix * SRC) This function copies the elements of the matrix SRC into the matrix DEST. The two matrices must have the same size. -- Function: int gsl_matrix_swap (gsl_matrix * M1, gsl_matrix * M2) This function exchanges the elements of the matrices M1 and M2 by copying. The two matrices must have the same size.  File: gsl-ref.info, Node: Copying rows and columns, Next: Exchanging rows and columns, Prev: Copying matrices, Up: Matrices 8.4.8 Copying rows and columns ------------------------------ The functions described in this section copy a row or column of a matrix into a vector. This allows the elements of the vector and the matrix to be modified independently. Note that if the matrix and the vector point to overlapping regions of memory then the result will be undefined. The same effect can be achieved with more generality using `gsl_vector_memcpy' with vector views of rows and columns. -- Function: int gsl_matrix_get_row (gsl_vector * V, const gsl_matrix * M, size_t I) This function copies the elements of the I-th row of the matrix M into the vector V. The length of the vector must be the same as the length of the row. -- Function: int gsl_matrix_get_col (gsl_vector * V, const gsl_matrix * M, size_t J) This function copies the elements of the J-th column of the matrix M into the vector V. The length of the vector must be the same as the length of the column. -- Function: int gsl_matrix_set_row (gsl_matrix * M, size_t I, const gsl_vector * V) This function copies the elements of the vector V into the I-th row of the matrix M. The length of the vector must be the same as the length of the row. -- Function: int gsl_matrix_set_col (gsl_matrix * M, size_t J, const gsl_vector * V) This function copies the elements of the vector V into the J-th column of the matrix M. The length of the vector must be the same as the length of the column.  File: gsl-ref.info, Node: Exchanging rows and columns, Next: Matrix operations, Prev: Copying rows and columns, Up: Matrices 8.4.9 Exchanging rows and columns --------------------------------- The following functions can be used to exchange the rows and columns of a matrix. -- Function: int gsl_matrix_swap_rows (gsl_matrix * M, size_t I, size_t J) This function exchanges the I-th and J-th rows of the matrix M in-place. -- Function: int gsl_matrix_swap_columns (gsl_matrix * M, size_t I, size_t J) This function exchanges the I-th and J-th columns of the matrix M in-place. -- Function: int gsl_matrix_swap_rowcol (gsl_matrix * M, size_t I, size_t J) This function exchanges the I-th row and J-th column of the matrix M in-place. The matrix must be square for this operation to be possible. -- Function: int gsl_matrix_transpose_memcpy (gsl_matrix * DEST, const gsl_matrix * SRC) This function makes the matrix DEST the transpose of the matrix SRC by copying the elements of SRC into DEST. This function works for all matrices provided that the dimensions of the matrix DEST match the transposed dimensions of the matrix SRC. -- Function: int gsl_matrix_transpose (gsl_matrix * M) This function replaces the matrix M by its transpose by copying the elements of the matrix in-place. The matrix must be square for this operation to be possible.  File: gsl-ref.info, Node: Matrix operations, Next: Finding maximum and minimum elements of matrices, Prev: Exchanging rows and columns, Up: Matrices 8.4.10 Matrix operations ------------------------ The following operations are defined for real and complex matrices. -- Function: int gsl_matrix_add (gsl_matrix * A, const gsl_matrix * B) This function adds the elements of matrix B to the elements of matrix A, a'(i,j) = a(i,j) + b(i,j). The two matrices must have the same dimensions. -- Function: int gsl_matrix_sub (gsl_matrix * A, const gsl_matrix * B) This function subtracts the elements of matrix B from the elements of matrix A, a'(i,j) = a(i,j) - b(i,j). The two matrices must have the same dimensions. -- Function: int gsl_matrix_mul_elements (gsl_matrix * A, const gsl_matrix * B) This function multiplies the elements of matrix A by the elements of matrix B, a'(i,j) = a(i,j) * b(i,j). The two matrices must have the same dimensions. -- Function: int gsl_matrix_div_elements (gsl_matrix * A, const gsl_matrix * B) This function divides the elements of matrix A by the elements of matrix B, a'(i,j) = a(i,j) / b(i,j). The two matrices must have the same dimensions. -- Function: int gsl_matrix_scale (gsl_matrix * A, const double X) This function multiplies the elements of matrix A by the constant factor X, a'(i,j) = x a(i,j). -- Function: int gsl_matrix_add_constant (gsl_matrix * A, const double X) This function adds the constant value X to the elements of the matrix A, a'(i,j) = a(i,j) + x.  File: gsl-ref.info, Node: Finding maximum and minimum elements of matrices, Next: Matrix properties, Prev: Matrix operations, Up: Matrices 8.4.11 Finding maximum and minimum elements of matrices ------------------------------------------------------- The following operations are only defined for real matrices. -- Function: double gsl_matrix_max (const gsl_matrix * M) This function returns the maximum value in the matrix M. -- Function: double gsl_matrix_min (const gsl_matrix * M) This function returns the minimum value in the matrix M. -- Function: void gsl_matrix_minmax (const gsl_matrix * M, double * MIN_OUT, double * MAX_OUT) This function returns the minimum and maximum values in the matrix M, storing them in MIN_OUT and MAX_OUT. -- Function: void gsl_matrix_max_index (const gsl_matrix * M, size_t * IMAX, size_t * JMAX) This function returns the indices of the maximum value in the matrix M, storing them in IMAX and JMAX. When there are several equal maximum elements then the first element found is returned, searching in row-major order. -- Function: void gsl_matrix_min_index (const gsl_matrix * M, size_t * IMIN, size_t * JMIN) This function returns the indices of the minimum value in the matrix M, storing them in IMIN and JMIN. When there are several equal minimum elements then the first element found is returned, searching in row-major order. -- Function: void gsl_matrix_minmax_index (const gsl_matrix * M, size_t * IMIN, size_t * JMIN, size_t * IMAX, size_t * JMAX) This function returns the indices of the minimum and maximum values in the matrix M, storing them in (IMIN,JMIN) and (IMAX,JMAX). When there are several equal minimum or maximum elements then the first elements found are returned, searching in row-major order.  File: gsl-ref.info, Node: Matrix properties, Next: Example programs for matrices, Prev: Finding maximum and minimum elements of matrices, Up: Matrices 8.4.12 Matrix properties ------------------------ -- Function: int gsl_matrix_isnull (const gsl_matrix * M) -- Function: int gsl_matrix_ispos (const gsl_matrix * M) -- Function: int gsl_matrix_isneg (const gsl_matrix * M) These functions return 1 if all the elements of the matrix M are zero, strictly positive, or strictly negative respectively, and 0 otherwise. To test for a non-negative matrix, use the expression `!gsl_matrix_isneg(m)'. To test whether a matrix is positive-definite, use the Cholesky decomposition (*note Cholesky Decomposition::).  File: gsl-ref.info, Node: Example programs for matrices, Prev: Matrix properties, Up: Matrices 8.4.13 Example programs for matrices ------------------------------------ The program below shows how to allocate, initialize and read from a matrix using the functions `gsl_matrix_alloc', `gsl_matrix_set' and `gsl_matrix_get'. #include #include 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 #include 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 #include #include #include int main (void) { size_t i,j; gsl_matrix *m = gsl_matrix_alloc (10, 10); for (i = 0; i < 10; i++) for (j = 0; j < 10; j++) gsl_matrix_set (m, i, j, sin (i) + cos (j)); for (j = 0; j < 10; j++) { gsl_vector_view column = gsl_matrix_column (m, j); double d; d = gsl_blas_dnrm2 (&column.vector); printf ("matrix column %d, norm = %g\n", j, d); } gsl_matrix_free (m); return 0; } Here is the output of the program, $ ./a.out matrix column 0, norm = 4.31461 matrix column 1, norm = 3.1205 matrix column 2, norm = 2.19316 matrix column 3, norm = 3.26114 matrix column 4, norm = 2.53416 matrix column 5, norm = 2.57281 matrix column 6, norm = 4.20469 matrix column 7, norm = 3.65202 matrix column 8, norm = 2.08524 matrix column 9, norm = 3.07313 The results can be confirmed using GNU OCTAVE, $ octave GNU Octave, version 2.0.16.92 octave> m = sin(0:9)' * ones(1,10) + ones(10,1) * cos(0:9); octave> sqrt(sum(m.^2)) ans = 4.3146 3.1205 2.1932 3.2611 2.5342 2.5728 4.2047 3.6520 2.0852 3.0731  File: gsl-ref.info, Node: Vector and Matrix References and Further Reading, Prev: Matrices, Up: Vectors and Matrices 8.5 References and Further Reading ================================== The block, vector and matrix objects in GSL follow the `valarray' model of C++. A description of this model can be found in the following reference, B. Stroustrup, `The C++ Programming Language' (3rd Ed), Section 22.4 Vector Arithmetic. Addison-Wesley 1997, ISBN 0-201-88954-4.  File: gsl-ref.info, Node: Permutations, Next: Combinations, Prev: Vectors and Matrices, Up: Top 9 Permutations ************** This chapter describes functions for creating and manipulating permutations. A permutation p is represented by an array of n integers in the range 0 to n-1, where each value p_i occurs once and only once. The application of a permutation p to a vector v yields a new vector v' where v'_i = v_{p_i}. For example, the array (0,1,3,2) represents a permutation which exchanges the last two elements of a four element vector. The corresponding identity permutation is (0,1,2,3). Note that the permutations produced by the linear algebra routines correspond to the exchange of matrix columns, and so should be considered as applying to row-vectors in the form v' = v P rather than column-vectors, when permuting the elements of a vector. The functions described in this chapter are defined in the header file `gsl_permutation.h'. * Menu: * The Permutation struct:: * Permutation allocation:: * Accessing permutation elements:: * Permutation properties:: * Permutation functions:: * Applying Permutations:: * Reading and writing permutations:: * Permutations in cyclic form:: * Permutation Examples:: * Permutation References and Further Reading::  File: gsl-ref.info, Node: The Permutation struct, Next: Permutation allocation, Up: Permutations 9.1 The Permutation struct ========================== A permutation is defined by a structure containing two components, the size of the permutation and a pointer to the permutation array. The elements of the permutation array are all of type `size_t'. The `gsl_permutation' structure looks like this, typedef struct { size_t size; size_t * data; } gsl_permutation;  File: gsl-ref.info, Node: Permutation allocation, Next: Accessing permutation elements, Prev: The Permutation struct, Up: Permutations 9.2 Permutation allocation ========================== -- Function: gsl_permutation * gsl_permutation_alloc (size_t N) This function allocates memory for a new permutation of size N. The permutation is not initialized and its elements are undefined. Use the function `gsl_permutation_calloc' if you want to create a permutation which is initialized to the identity. A null pointer is returned if insufficient memory is available to create the permutation. -- Function: gsl_permutation * gsl_permutation_calloc (size_t N) This function allocates memory for a new permutation of size N and initializes it to the identity. A null pointer is returned if insufficient memory is available to create the permutation. -- Function: void gsl_permutation_init (gsl_permutation * P) This function initializes the permutation P to the identity, i.e. (0,1,2,...,n-1). -- Function: void gsl_permutation_free (gsl_permutation * P) This function frees all the memory used by the permutation P. -- Function: int gsl_permutation_memcpy (gsl_permutation * DEST, const gsl_permutation * SRC) This function copies the elements of the permutation SRC into the permutation DEST. The two permutations must have the same size.  File: gsl-ref.info, Node: Accessing permutation elements, Next: Permutation properties, Prev: Permutation allocation, Up: Permutations 9.3 Accessing permutation elements ================================== The following functions can be used to access and manipulate permutations. -- Function: size_t gsl_permutation_get (const gsl_permutation * P, const size_t I) This function returns the value of the I-th element of the permutation P. If I lies outside the allowed range of 0 to N-1 then the error handler is invoked and 0 is returned. -- Function: int gsl_permutation_swap (gsl_permutation * P, const size_t I, const size_t J) This function exchanges the I-th and J-th elements of the permutation P.  File: gsl-ref.info, Node: Permutation properties, Next: Permutation functions, Prev: Accessing permutation elements, Up: Permutations 9.4 Permutation properties ========================== -- Function: size_t gsl_permutation_size (const gsl_permutation * P) This function returns the size of the permutation P. -- Function: size_t * gsl_permutation_data (const gsl_permutation * P) This function returns a pointer to the array of elements in the permutation P. -- Function: int gsl_permutation_valid (gsl_permutation * P) This function checks that the permutation P is valid. The N elements should contain each of the numbers 0 to N-1 once and only once.  File: gsl-ref.info, Node: Permutation functions, Next: Applying Permutations, Prev: Permutation properties, Up: Permutations 9.5 Permutation functions ========================= -- Function: void gsl_permutation_reverse (gsl_permutation * P) This function reverses the elements of the permutation P. -- Function: int gsl_permutation_inverse (gsl_permutation * INV, const gsl_permutation * P) This function computes the inverse of the permutation P, storing the result in INV. -- Function: int gsl_permutation_next (gsl_permutation * P) This function advances the permutation P to the next permutation in lexicographic order and returns `GSL_SUCCESS'. If no further permutations are available it returns `GSL_FAILURE' and leaves P unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order. -- Function: int gsl_permutation_prev (gsl_permutation * P) This function steps backwards from the permutation P to the previous permutation in lexicographic order, returning `GSL_SUCCESS'. If no previous permutation is available it returns `GSL_FAILURE' and leaves P unmodified.  File: gsl-ref.info, Node: Applying Permutations, Next: Reading and writing permutations, Prev: Permutation functions, Up: Permutations 9.6 Applying Permutations ========================= -- Function: int gsl_permute (const size_t * P, double * DATA, size_t STRIDE, size_t N) This function applies the permutation P to the array DATA of size N with stride STRIDE. -- Function: int gsl_permute_inverse (const size_t * P, double * DATA, size_t STRIDE, size_t N) This function applies the inverse of the permutation P to the array DATA of size N with stride STRIDE. -- Function: int gsl_permute_vector (const gsl_permutation * P, gsl_vector * V) This function applies the permutation P to the elements of the vector V, considered as a row-vector acted on by a permutation matrix from the right, v' = v P. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation P and the vector V must have the same length. -- Function: int gsl_permute_vector_inverse (const gsl_permutation * P, gsl_vector * V) This function applies the inverse of the permutation P to the elements of the vector V, considered as a row-vector acted on by an inverse permutation matrix from the right, v' = v P^T. Note that for permutation matrices the inverse is the same as the transpose. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation P and the vector V must have the same length. -- Function: int gsl_permutation_mul (gsl_permutation * P, const gsl_permutation * PA, const gsl_permutation * PB) This function combines the two permutations PA and PB into a single permutation P, where p = pa . pb. The permutation P is equivalent to applying pb first and then PA.  File: gsl-ref.info, Node: Reading and writing permutations, Next: Permutations in cyclic form, Prev: Applying Permutations, Up: Permutations 9.7 Reading and writing permutations ==================================== The library provides functions for reading and writing permutations to a file as binary data or formatted text. -- Function: int gsl_permutation_fwrite (FILE * STREAM, const gsl_permutation * P) This function writes the elements of the permutation P to the stream STREAM in binary format. The function returns `GSL_EFAILED' if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures. -- Function: int gsl_permutation_fread (FILE * STREAM, gsl_permutation * P) This function reads into the permutation P from the open stream STREAM in binary format. The permutation P must be preallocated with the correct length since the function uses the size of P to determine how many bytes to read. The function returns `GSL_EFAILED' if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture. -- Function: int gsl_permutation_fprintf (FILE * STREAM, const gsl_permutation * P, const char * FORMAT) This function writes the elements of the permutation P line-by-line to the stream STREAM using the format specifier FORMAT, which should be suitable for a type of SIZE_T. On a GNU system the type modifier `Z' represents `size_t', so `"%Zu\n"' is a suitable format. The function returns `GSL_EFAILED' if there was a problem writing to the file. -- Function: int gsl_permutation_fscanf (FILE * STREAM, gsl_permutation * P) This function reads formatted data from the stream STREAM into the permutation P. The permutation P must be preallocated with the correct length since the function uses the size of P to determine how many numbers to read. The function returns `GSL_EFAILED' if there was a problem reading from the file.  File: gsl-ref.info, Node: Permutations in cyclic form, Next: Permutation Examples, Prev: Reading and writing permutations, Up: Permutations 9.8 Permutations in cyclic form =============================== A permutation can be represented in both "linear" and "cyclic" notations. The functions described in this section convert between the two forms. The linear notation is an index mapping, and has already been described above. The cyclic notation expresses a permutation as a series of circular rearrangements of groups of elements, or "cycles". For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different sets of elements can be combined independently, for example (1 2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange of elements 4 and 5. A cycle of length one represents an element which is unchanged by the permutation and is referred to as a "singleton". It can be shown that every permutation can be decomposed into combinations of cycles. The decomposition is not unique, but can always be rearranged into a standard "canonical form" by a reordering of elements. The library uses the canonical form defined in Knuth's `Art of Computer Programming' (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178. The procedure for obtaining the canonical form given by Knuth is, 1. Write all singleton cycles explicitly 2. Within each cycle, put the smallest number first 3. Order the cycles in decreasing order of the first number in the cycle. For example, the linear representation (2 4 3 0 1) is represented as (1 4) (0 2 3) in canonical form. The permutation corresponds to an exchange of elements 1 and 4, and rotation of elements 0, 2 and 3. The important property of the canonical form is that it can be reconstructed from the contents of each cycle without the brackets. In addition, by removing the brackets it can be considered as a linear representation of a different permutation. In the example given above the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has many applications in the theory of permutations. -- Function: int gsl_permutation_linear_to_canonical (gsl_permutation * Q, const gsl_permutation * P) This function computes the canonical form of the permutation P and stores it in the output argument Q. -- Function: int gsl_permutation_canonical_to_linear (gsl_permutation * P, const gsl_permutation * Q) This function converts a permutation Q in canonical form back into linear form storing it in the output argument P. -- Function: size_t gsl_permutation_inversions (const gsl_permutation * P) This function counts the number of inversions in the permutation P. An inversion is any pair of elements that are not in order. For example, the permutation 2031 has three inversions, corresponding to the pairs (2,0) (2,1) and (3,1). The identity permutation has no inversions. -- Function: size_t gsl_permutation_linear_cycles (const gsl_permutation * P) This function counts the number of cycles in the permutation P, given in linear form. -- Function: size_t gsl_permutation_canonical_cycles (const gsl_permutation * Q) This function counts the number of cycles in the permutation Q, given in canonical form.  File: gsl-ref.info, Node: Permutation Examples, Next: Permutation References and Further Reading, Prev: Permutations in cyclic form, Up: Permutations 9.9 Examples ============ The example program below creates a random permutation (by shuffling the elements of the identity) and finds its inverse. #include #include #include #include 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 #include int main (void) { gsl_permutation * p = gsl_permutation_alloc (3); gsl_permutation_init (p); do { gsl_permutation_fprintf (stdout, p, " %u"); printf ("\n"); } while (gsl_permutation_next(p) == GSL_SUCCESS); gsl_permutation_free (p); return 0; } Here is the output from the program, $ ./a.out 0 1 2 0 2 1 1 0 2 1 2 0 2 0 1 2 1 0 The permutations are generated in lexicographic order. To reverse the sequence, begin with the final permutation (which is the reverse of the identity) and replace `gsl_permutation_next' with `gsl_permutation_prev'.  File: gsl-ref.info, Node: Permutation References and Further Reading, Prev: Permutation Examples, Up: Permutations 9.10 References and Further Reading =================================== The subject of permutations is covered extensively in Knuth's `Sorting and Searching', Donald E. Knuth, `The Art of Computer Programming: Sorting and Searching' (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850. For the definition of the "canonical form" see, Donald E. Knuth, `The Art of Computer Programming: Fundamental Algorithms' (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850. Section 1.3.3, `An Unusual Correspondence', p.178-179.  File: gsl-ref.info, Node: Combinations, Next: Sorting, Prev: Permutations, Up: Top 10 Combinations *************** This chapter describes functions for creating and manipulating combinations. A combination c is represented by an array of k integers in the range 0 to n-1, where each value c_i occurs at most once. The combination c corresponds to indices of k elements chosen from an n element vector. Combinations are useful for iterating over all k-element subsets of a set. The functions described in this chapter are defined in the header file `gsl_combination.h'. * Menu: * The Combination struct:: * Combination allocation:: * Accessing combination elements:: * Combination properties:: * Combination functions:: * Reading and writing combinations:: * Combination Examples:: * Combination References and Further Reading::  File: gsl-ref.info, Node: The Combination struct, Next: Combination allocation, Up: Combinations 10.1 The Combination struct =========================== A combination is defined by a structure containing three components, the values of n and k, and a pointer to the combination array. The elements of the combination array are all of type `size_t', and are stored in increasing order. The `gsl_combination' structure looks like this, typedef struct { size_t n; size_t k; size_t *data; } gsl_combination;  File: gsl-ref.info, Node: Combination allocation, Next: Accessing combination elements, Prev: The Combination struct, Up: Combinations 10.2 Combination allocation =========================== -- Function: gsl_combination * gsl_combination_alloc (size_t N, size_t K) This function allocates memory for a new combination with parameters N, K. The combination is not initialized and its elements are undefined. Use the function `gsl_combination_calloc' if you want to create a combination which is initialized to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination. -- Function: gsl_combination * gsl_combination_calloc (size_t N, size_t K) This function allocates memory for a new combination with parameters N, K and initializes it to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination. -- Function: void gsl_combination_init_first (gsl_combination * C) This function initializes the combination C to the lexicographically first combination, i.e. (0,1,2,...,k-1). -- Function: void gsl_combination_init_last (gsl_combination * C) This function initializes the combination C to the lexicographically last combination, i.e. (n-k,n-k+1,...,n-1). -- Function: void gsl_combination_free (gsl_combination * C) This function frees all the memory used by the combination C. -- Function: int gsl_combination_memcpy (gsl_combination * DEST, const gsl_combination * SRC) This function copies the elements of the combination SRC into the combination DEST. The two combinations must have the same size.  File: gsl-ref.info, Node: Accessing combination elements, Next: Combination properties, Prev: Combination allocation, Up: Combinations 10.3 Accessing combination elements =================================== The following function can be used to access the elements of a combination. -- Function: size_t gsl_combination_get (const gsl_combination * C, const size_t I) This function returns the value of the I-th element of the combination C. If I lies outside the allowed range of 0 to K-1 then the error handler is invoked and 0 is returned.  File: gsl-ref.info, Node: Combination properties, Next: Combination functions, Prev: Accessing combination elements, Up: Combinations 10.4 Combination properties =========================== -- Function: size_t gsl_combination_n (const gsl_combination * C) This function returns the range (n) of the combination C. -- Function: size_t gsl_combination_k (const gsl_combination * C) This function returns the number of elements (k) in the combination C. -- Function: size_t * gsl_combination_data (const gsl_combination * C) This function returns a pointer to the array of elements in the combination C. -- Function: int gsl_combination_valid (gsl_combination * C) This function checks that the combination C is valid. The K elements should lie in the range 0 to N-1, with each value occurring once at most and in increasing order.  File: gsl-ref.info, Node: Combination functions, Next: Reading and writing combinations, Prev: Combination properties, Up: Combinations 10.5 Combination functions ========================== -- Function: int gsl_combination_next (gsl_combination * C) This function advances the combination C to the next combination in lexicographic order and returns `GSL_SUCCESS'. If no further combinations are available it returns `GSL_FAILURE' and leaves C unmodified. Starting with the first combination and repeatedly applying this function will iterate through all possible combinations of a given order. -- Function: int gsl_combination_prev (gsl_combination * C) This function steps backwards from the combination C to the previous combination in lexicographic order, returning `GSL_SUCCESS'. If no previous combination is available it returns `GSL_FAILURE' and leaves C unmodified.  File: gsl-ref.info, Node: Reading and writing combinations, Next: Combination Examples, Prev: Combination functions, Up: Combinations 10.6 Reading and writing combinations ===================================== The library provides functions for reading and writing combinations to a file as binary data or formatted text. -- Function: int gsl_combination_fwrite (FILE * STREAM, const gsl_combination * C) This function writes the elements of the combination C to the stream STREAM in binary format. The function returns `GSL_EFAILED' if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures. -- Function: int gsl_combination_fread (FILE * STREAM, gsl_combination * C) This function reads elements from the open stream STREAM into the combination C in binary format. The combination C must be preallocated with correct values of n and k since the function uses the size of C to determine how many bytes to read. The function returns `GSL_EFAILED' if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture. -- Function: int gsl_combination_fprintf (FILE * STREAM, const gsl_combination * C, const char * FORMAT) This function writes the elements of the combination C line-by-line to the stream STREAM using the format specifier FORMAT, which should be suitable for a type of SIZE_T. On a GNU system the type modifier `Z' represents `size_t', so `"%Zu\n"' is a suitable format. The function returns `GSL_EFAILED' if there was a problem writing to the file. -- Function: int gsl_combination_fscanf (FILE * STREAM, gsl_combination * C) This function reads formatted data from the stream STREAM into the combination C. The combination C must be preallocated with correct values of n and k since the function uses the size of C to determine how many numbers to read. The function returns `GSL_EFAILED' if there was a problem reading from the file.  File: gsl-ref.info, Node: Combination Examples, Next: Combination References and Further Reading, Prev: Reading and writing combinations, Up: Combinations 10.7 Examples ============= The example program below prints all subsets of the set {0,1,2,3} ordered by size. Subsets of the same size are ordered lexicographically. #include #include int main (void) { gsl_combination * c; size_t i; printf ("All subsets of {0,1,2,3} by size:\n") ; for (i = 0; i <= 4; i++) { c = gsl_combination_calloc (4, i); do { printf ("{"); gsl_combination_fprintf (stdout, c, " %u"); printf (" }\n"); } while (gsl_combination_next (c) == GSL_SUCCESS); gsl_combination_free (c); } return 0; } Here is the output from the program, $ ./a.out All subsets of {0,1,2,3} by size: { } { 0 } { 1 } { 2 } { 3 } { 0 1 } { 0 2 } { 0 3 } { 1 2 } { 1 3 } { 2 3 } { 0 1 2 } { 0 1 3 } { 0 2 3 } { 1 2 3 } { 0 1 2 3 } All 16 subsets are generated, and the subsets of each size are sorted lexicographically.  File: gsl-ref.info, Node: Combination References and Further Reading, Prev: Combination Examples, Up: Combinations 10.8 References and Further Reading =================================== Further information on combinations can be found in, Donald L. Kreher, Douglas R. Stinson, `Combinatorial Algorithms: Generation, Enumeration and Search', 1998, CRC Press LLC, ISBN 084933988X  File: gsl-ref.info, Node: Sorting, Next: BLAS Support, Prev: Combinations, Up: Top 11 Sorting ********** This chapter describes functions for sorting data, both directly and indirectly (using an index). All the functions use the "heapsort" algorithm. Heapsort is an O(N \log N) algorithm which operates in-place and does not require any additional storage. It also provides consistent performance, the running time for its worst-case (ordered data) being not significantly longer than the average and best cases. Note that the heapsort algorithm does not preserve the relative ordering of equal elements--it is an "unstable" sort. However the resulting order of equal elements will be consistent across different platforms when using these functions. * Menu: * Sorting objects:: * Sorting vectors:: * Selecting the k smallest or largest elements:: * Computing the rank:: * Sorting Examples:: * Sorting References and Further Reading::  File: gsl-ref.info, Node: Sorting objects, Next: Sorting vectors, Up: Sorting 11.1 Sorting objects ==================== The following function provides a simple alternative to the standard library function `qsort'. It is intended for systems lacking `qsort', not as a replacement for it. The function `qsort' should be used whenever possible, as it will be faster and can provide stable ordering of equal elements. Documentation for `qsort' is available in the `GNU C Library Reference Manual'. The functions described in this section are defined in the header file `gsl_heapsort.h'. -- Function: void gsl_heapsort (void * ARRAY, size_t COUNT, size_t SIZE, gsl_comparison_fn_t COMPARE) This function sorts the COUNT elements of the array ARRAY, each of size SIZE, into ascending order using the comparison function COMPARE. The type of the comparison function is defined by, int (*gsl_comparison_fn_t) (const void * a, const void * b) A comparison function should return a negative integer if the first argument is less than the second argument, `0' if the two arguments are equal and a positive integer if the first argument is greater than the second argument. For example, the following function can be used to sort doubles into ascending numerical order. int compare_doubles (const double * a, const double * b) { if (*a > *b) return 1; else if (*a < *b) return -1; else return 0; } The appropriate function call to perform the sort is, gsl_heapsort (array, count, sizeof(double), compare_doubles); Note that unlike `qsort' the heapsort algorithm cannot be made into a stable sort by pointer arithmetic. The trick of comparing pointers for equal elements in the comparison function does not work for the heapsort algorithm. The heapsort algorithm performs an internal rearrangement of the data which destroys its initial ordering. -- Function: int gsl_heapsort_index (size_t * P, const void * ARRAY, size_t COUNT, size_t SIZE, gsl_comparison_fn_t COMPARE) This function indirectly sorts the COUNT elements of the array ARRAY, each of size SIZE, into ascending order using the comparison function COMPARE. The resulting permutation is stored in P, an array of length N. The elements of P give the index of the array element which would have been stored in that position if the array had been sorted in place. The first element of P gives the index of the least element in ARRAY, and the last element of P gives the index of the greatest element in ARRAY. The array itself is not changed.  File: gsl-ref.info, Node: Sorting vectors, Next: Selecting the k smallest or largest elements, Prev: Sorting objects, Up: Sorting 11.2 Sorting vectors ==================== The following functions will sort the elements of an array or vector, either directly or indirectly. They are defined for all real and integer types using the normal suffix rules. For example, the `float' versions of the array functions are `gsl_sort_float' and `gsl_sort_float_index'. The corresponding vector functions are `gsl_sort_vector_float' and `gsl_sort_vector_float_index'. The prototypes are available in the header files `gsl_sort_float.h' `gsl_sort_vector_float.h'. The complete set of prototypes can be included using the header files `gsl_sort.h' and `gsl_sort_vector.h'. There are no functions for sorting complex arrays or vectors, since the ordering of complex numbers is not uniquely defined. To sort a complex vector by magnitude compute a real vector containing the magnitudes of the complex elements, and sort this vector indirectly. The resulting index gives the appropriate ordering of the original complex vector. -- Function: void gsl_sort (double * DATA, size_t STRIDE, size_t N) This function sorts the N elements of the array DATA with stride STRIDE into ascending numerical order. -- Function: void gsl_sort_vector (gsl_vector * V) This function sorts the elements of the vector V into ascending numerical order. -- Function: 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.