GNU Scientific Library
Reference Manual
Edition 1.10, for GSL Version 1.10
10 September 2007
Mark Galassi
Los Alamos National Laboratory
Jim Davies
Department of Computer Science, Georgia Institute of Technology
James Theiler
Astrophysics and Radiation Measurements Group, Los Alamos National Laboratory
Brian Gough
Network Theory Limited
Gerard Jungman
Theoretical Astrophysics Group, Los Alamos National Laboratory
Michael Booth
Department of Physics and Astronomy, The Johns Hopkins University
Fabrice Rossi
University of Paris-Dauphine
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.” 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.
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Table of Contents
1 Introduction ...... .... ..... ..... .... ..... .. 1
1.1 Routines available in GSL.................................... 1
1.2 GSL is Free Software ........................................ 1
1.3 Obtaining GSL.............................................. 2
1.4 No Warranty ................................................ 2
1.5 Reporting Bugs ............................................. 3
1.6 Further Information ......................................... 3
1.7 Conventions used in this manual.............................. 3
2 Using the library ...... ..... ..... .... ..... . 4
2.1 An Example Program ....................................... 4
2.2 Compiling and Linking ...................................... 4
2.2.1 Linking programs with the library........................ 4
2.2.2 Linking with an alternative BLAS library ................. 5
2.3 Shared Libraries............................................. 5
2.4 ANSI C Compliance ......................................... 6
2.5 Inline functions ............................................. 6
2.6 Long double ................................................ 6
2.7 Portability functions ......................................... 7
2.8 Alternative optimized functions............................... 7
2.9 Support for different numeric types ........................... 8
2.10 Compatibility with C++ ..................................... 9
2.11 Aliasing of arrays .......................................... 9
2.12 Thread-safety .............................................. 9
2.13 Deprecated Functions ....................................... 9
2.14 Code Reuse ............................................... 10
3 Error Handling ... ..... ..... ........... ... 11
3.1 Error Reporting ............................................ 11
3.2 Error Codes ............................................... 11
3.3 Error Handlers ............................................. 12
3.4 Using GSL error reporting in your own functions ............. 13
3.5 Examples .................................................. 14
4 Mathematical Functions......... ..... ..... 16
4.1 Mathematical Constants .................................... 16
4.2 Infinities and Not-a-number ................................. 16
4.3 Elementary Functions ...................................... 17
4.4 Small integer powers........................................ 18
4.5 Testing the Sign of Numbers ................................ 18
4.6 Testing for Odd and Even Numbers.......................... 18
4.7 Maximum and Minimum functions........................... 19
4.8 Approximate Comparison of Floating Point Numbers ......... 19
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5 Complex Numbers .. ..... .... ..... ........ 21
5.1 Complex numbers .......................................... 21
5.2 Properties of complex numbers .............................. 22
5.3 Complex arithmetic operators ............................... 22
5.4 Elementary Complex Functions .............................. 23
5.5 Complex Trigonometric Functions ........................... 24
5.6 Inverse Complex Trigonometric Functions .................... 24
5.7 Complex Hyperbolic Functions .............................. 25
5.8 Inverse Complex Hyperbolic Functions ....................... 26
5.9 References and Further Reading ............................. 26
6 Polynomials .... ........... ..... .... ..... . 28
6.1 Polynomial Evaluation ...................................... 28
6.2 Divided Difference Representation of Polynomials ............. 28
6.3 Quadratic Equations ....................................... 28
6.4 Cubic Equations ........................................... 29
6.5 General Polynomial Equations .............................. 30
6.6 Examples .................................................. 30
6.7 References and Further Reading ............................. 31
7 Special Functions ............. ..... .... ... 32
7.1 Usage ..................................................... 32
7.2 The gsl
sf result struct ..................................... 32
7.3 Modes ..................................................... 33
7.4 Airy Functions and Derivatives .............................. 33
7.4.1 Airy Functions ........................................ 33
7.4.2 Derivatives of Airy Functions ........................... 34
7.4.3 Zeros of Airy Functions ................................ 34
7.4.4 Zeros of Derivatives of Airy Functions ................... 35
7.5 Bessel Functions ........................................... 35
7.5.1 Regular Cylindrical Bessel Functions .................... 35
7.5.2 Irregular Cylindrical Bessel Functions ................... 35
7.5.3 Regular Modified Cylindrical Bessel Functions ........... 36
7.5.4 Irregular Modified Cylindrical Bessel Functions .......... 37
7.5.5 Regular Spherical Bessel Functions ...................... 38
7.5.6 Irregular Spherical Bessel Functions ..................... 39
7.5.7 Regular Modified Spherical Bessel Functions ............. 39
7.5.8 Irregular Modified Spherical Bessel Functions ............ 40
7.5.9 Regular Bessel Function—Fractional Order .............. 40
7.5.10 Irregular Bessel Functions—Fractional Order............ 41
7.5.11 Regular Modified Bessel Functions—Fractional Order .... 41
7.5.12 Irregular Modified Bessel Functions—Fractional Order ... 41
7.5.13 Zeros of Regular Bessel Functions ...................... 42
7.6 Clausen Functions .......................................... 42
7.7 Coulomb Functions ......................................... 42
7.7.1 Normalized Hydrogenic Bound States ................... 42
7.7.2 Coulomb Wave Functions............................... 43
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7.7.3 Coulomb Wave Function Normalization Constant ........ 44
7.8 Coupling Coefficients ....................................... 44
7.8.1 3-j Symbols ........................................... 44
7.8.2 6-j Symbols ........................................... 45
7.8.3 9-j Symbols ........................................... 45
7.9 Dawson Function ........................................... 45
7.10 Debye Functions .......................................... 45
7.11 Dilogarithm............................................... 46
7.11.1 Real Argument ....................................... 46
7.11.2 Complex Argument ................................... 46
7.12 Elementary Operations .................................... 46
7.13 Elliptic Integrals .......................................... 47
7.13.1 Definition of Legendre Forms .......................... 47
7.13.2 Definition of Carlson Forms ........................... 47
7.13.3 Legendre Form of Complete Elliptic Integrals ........... 47
7.13.4 Legendre Form of Incomplete Elliptic Integrals .......... 48
7.13.5 Carlson Forms ........................................ 49
7.14 Elliptic Functions (Jacobi) ................................. 49
7.15 Error Functions ........................................... 50
7.15.1 Error Function ....................................... 50
7.15.2 Complementary Error Function ........................ 50
7.15.3 Log Complementary Error Function .................... 50
7.15.4 Probability functions .................................. 50
7.16 Exponential Functions ..................................... 51
7.16.1 Exponential Function ................................. 51
7.16.2 Relative Exponential Functions ........................ 51
7.16.3 Exponentiation With Error Estimate ................... 52
7.17 Exponential Integrals ...................................... 52
7.17.1 Exponential Integral .................................. 52
7.17.2 Ei(x) ................................................ 53
7.17.3 Hyperbolic Integrals .................................. 53
7.17.4 Ei
3(x) .............................................. 53
7.17.5 Trigonometric Integrals ............................... 53
7.17.6 Arctangent Integral ................................... 53
7.18 Fermi-Dirac Function ...................................... 54
7.18.1 Complete Fermi-Dirac Integrals ........................ 54
7.18.2 Incomplete Fermi-Dirac Integrals ...................... 55
7.19 Gamma and Beta Functions................................ 55
7.19.1 Gamma Functions .................................... 55
7.19.2 Factorials ............................................ 56
7.19.3 Pochhammer Symbol ................................. 57
7.19.4 Incomplete Gamma Functions ......................... 57
7.19.5 Beta Functions ....................................... 58
7.19.6 Incomplete Beta Function ............................. 58
7.20 Gegenbauer Functions ..................................... 58
7.21 Hypergeometric Functions ................................. 59
7.22 Laguerre Functions ........................................ 60
7.23 Lambert W Functions ..................................... 61
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