GNU Scientific Library

Reference Manual

Edition 1.8, for GSL Version 1.8 30 March 2006

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 \begin{matrix}c \\ ◯ \end{matrix} 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006 The GSL Team. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with the Invariant Sections being “GNU General Public License” and “Free Software Needs Free Documentation”, the Front-Cover text being “A GNU Manual”, and with the Back-Cover Text being (a) (see below). A copy of the license is included in the section entitled “GNU Free Documentation License”.

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

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

5 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Properties of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3 Complex arithmetic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.4 Elementary Complex Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.5 Complex Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.6 Inverse Complex Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 23 5.7 Complex Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.8 Inverse Complex Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 25 5.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.1 Polynomial Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 Divided Difference Representation of Polynomials . . . . . . . . . . . . . 27 6.3 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.4 Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.5 General Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.7 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.2 The gsl sf result struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.3 Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.4 Airy Functions and Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.4.1 Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.4.2 Derivatives of Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7.4.3 Zeros of Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7.4.4 Zeros of Derivatives of Airy Functions . . . . . . . . . . . . . . . . . . . 34

7.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.5.1 Regular Cylindrical Bessel Functions . . . . . . . . . . . . . . . . . . . . 34 7.5.2 Irregular Cylindrical Bessel Functions . . . . . . . . . . . . . . . . . . . 34 7.5.3 Regular Modified Cylindrical Bessel Functions . . . . . . . . . . . 35 7.5.4 Irregular Modified Cylindrical Bessel Functions . . . . . . . . . . 36 7.5.5 Regular Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . 37 7.5.6 Irregular Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . 38 7.5.7 Regular Modified Spherical Bessel Functions . . . . . . . . . . . . . 38 7.5.8 Irregular Modified Spherical Bessel Functions . . . . . . . . . . . . 39 7.5.9 Regular Bessel Function—Fractional Order . . . . . . . . . . . . . . 39 7.5.10 Irregular Bessel Functions—Fractional Order. . . . . . . . . . . . 40 7.5.11 Regular Modified Bessel Functions—Fractional Order. … 40 7.5.12 Irregular Modified Bessel Functions—Fractional Order … 40 7.5.13 Zeros of Regular Bessel Functions . . . . . . . . . . . . . . . . . . . . . . 41

7.6 Clausen Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.7 Coulomb Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.7.1 Normalized Hydrogenic Bound States . . . . . . . . . . . . . . . . . . . 41 7.7.2 Coulomb Wave Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7.7.3 Coulomb Wave Function Normalization Constant . . . . . . . . 43 7.8 Coupling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.8.1 3-j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.8.2 6-j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.8.3 9-j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.9 Dawson Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.10 Debye Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.11 Dilogarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.11.1 Real Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.11.2 Complex Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.12 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.13 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.13.1 Definition of Legendre Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.13.2 Definition of Carlson Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.13.3 Legendre Form of Complete Elliptic Integrals . . . . . . . . . . . 46 7.13.4 Legendre Form of Incomplete Elliptic Integrals . . . . . . . . . . 47 7.13.5 Carlson Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.14 Elliptic Functions (Jacobi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.15 Error Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.15.1 Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.15.2 Complementary Error Function . . . . . . . . . . . . . . . . . . . . . . . . 48 7.15.3 Log Complementary Error Function . . . . . . . . . . . . . . . . . . . . 48 7.15.4 Probability functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.16 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.16.1 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.16.2 Relative Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 50 7.16.3 Exponentiation With Error Estimate . . . . . . . . . . . . . . . . . . . 50

7.17 Exponential Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.17.1 Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.17.2 Ei(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.17.3 Hyperbolic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.17.4 Ei 3(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.17.5 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.17.6 Arctangent Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7.18 Fermi-Dirac Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.18.1 Complete Fermi-Dirac Integrals . . . . . . . . . . . . . . . . . . . . . . . . 52 7.18.2 Incomplete Fermi-Dirac Integrals . . . . . . . . . . . . . . . . . . . . . . 53

7.19 Gamma and Beta Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.19.1 Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.19.2 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.19.3 Pochhammer Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.19.4 Incomplete Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.19.5 Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.19.6 Incomplete Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.20 Gegenbauer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.21 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.22 Laguerre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.23 Lambert W Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.24 Legendre Functions and Spherical Harmonics . . . . . . . . . . . . . . . . 60

7.24.1 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.24.2 Associated Legendre Polynomials and Spherical Harmonics

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.24.3 Conical Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.24.4 Radial Functions for Hyperbolic Space . . . . . . . . . . . . . . . . . 62

7.25 Logarithm and Related Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.26 Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.27 Psi (Digamma) Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.27.1 Digamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.27.2 Trigamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.27.3 Polygamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.28 Synchrotron Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.29 Transport Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.30 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.30.1 Circular Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 65 7.30.2 Trigonometric Functions for Complex Arguments. . . . . . . . 66 7.30.3 Hyperbolic Trigonometric Functions. . . . . . . . . . . . . . . . . . . . 66 7.30.4 Conversion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.30.5 Restriction Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.30.6 Trigonometric Functions With Error Estimates. . . . . . . . . . 67

7.31 Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.31.1 Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.31.2 Riemann Zeta Function Minus One . . . . . . . . . . . . . . . . . . . . 67 7.31.3 Hurwitz Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.31.4 Eta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.32 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.33 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8 Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . 70

8.1 Data types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2 Blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8.2.1 Block allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2.2 Reading and writing blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2.3 Example programs for blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.3.1 Vector allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.3.2 Accessing vector elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.3.3 Initializing vector elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.3.4 Reading and writing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.3.5 Vector views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.3.6 Copying vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.3.7 Exchanging elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.3.8 Vector operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.3.9 Finding maximum and minimum elements of vectors . . . . . 78 8.3.10 Vector properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.3.11 Example programs for vectors . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.4.1 Matrix allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.4.2 Accessing matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.4.3 Initializing matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.4.4 Reading and writing matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.4.5 Matrix views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.4.6 Creating row and column views . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.4.7 Copying matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.4.8 Copying rows and columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.4.9 Exchanging rows and columns . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.4.10 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.4.11 Finding maximum and minimum elements of matrices … 88 8.4.12 Matrix properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.4.13 Example programs for matrices . . . . . . . . . . . . . . . . . . . . . . . . 89

8.5 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.1 The Permutation struct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.2 Permutation allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.3 Accessing permutation elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.4 Permutation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.5 Permutation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.6 Applying Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.7 Reading and writing permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.8 Permutations in cyclic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 9.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.10 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . 100

10.1 The Combination struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10.2 Combination allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10.3 Accessing combination elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.4 Combination properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.5 Combination functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.6 Reading and writing combinations . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.7 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10.8 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11 Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

11.1 Sorting objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 11.2 Sorting vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11.3 Selecting the k smallest or largest elements . . . . . . . . . . . . . . . . . 106 11.4 Computing the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

12 BLAS Support . . . . . . . . . . . . . . . . . . . . . . . . . 110

12.1 GSL BLAS Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 12.1.1 Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 12.1.2 Level 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 12.1.3 Level 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

12.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 12.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

13 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 122

13.1 LU Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 13.2 QR Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 13.3 QR Decomposition with Column Pivoting . . . . . . . . . . . . . . . . . . 125 13.4 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 13.5 Cholesky Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 13.6 Tridiagonal Decomposition of Real Symmetric Matrices . . . . . 128 13.7 Tridiagonal Decomposition of Hermitian Matrices. . . . . . . . . . . 129 13.8 Bidiagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 13.9 Householder Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 13.10 Householder solver for linear systems . . . . . . . . . . . . . . . . . . . . . 131 13.11 Tridiagonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 13.12 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 13.13 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 134

14 Eigensystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

14.1 Real Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 14.2 Complex Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 14.3 Sorting Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 136 14.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 14.5 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

15 Fast Fourier Transforms (FFTs) . . . . . . . . . 140

15.1 Mathematical Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 15.2 Overview of complex data FFTs. . . . . . . . . . . . . . . . . . . . . . . . . . . 141 15.3 Radix-2 FFT routines for complex data . . . . . . . . . . . . . . . . . . . . 142 15.4 Mixed-radix FFT routines for complex data . . . . . . . . . . . . . . . . 144 15.5 Overview of real data FFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 15.6 Radix-2 FFT routines for real data . . . . . . . . . . . . . . . . . . . . . . . . 149 15.7 Mixed-radix FFT routines for real data . . . . . . . . . . . . . . . . . . . . 150 15.8 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

16 Numerical Integration . . . . . . . . . . . . . . . . . . 157

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 16.1.1 Integrands without weight functions. . . . . . . . . . . . . . . . . . . 158 16.1.2 Integrands with weight functions. . . . . . . . . . . . . . . . . . . . . . 158 16.1.3 Integrands with singular weight functions. . . . . . . . . . . . . . 158

16.2 QNG non-adaptive Gauss-Kronrod integration . . . . . . . . . . . . . 158 16.3 QAG adaptive integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 16.4 QAGS adaptive integration with singularities. . . . . . . . . . . . . . . 159 16.5 QAGP adaptive integration with known singular points . . . . . 160 16.6 QAGI adaptive integration on infinite intervals . . . . . . . . . . . . . 160 16.7 QAWC adaptive integration for Cauchy principal values . . . . . 161 16.8 QAWS adaptive integration for singular functions. . . . . . . . . . . 161 16.9 QAWO adaptive integration for oscillatory functions . . . . . . . . 163 16.10 QAWF adaptive integration for Fourier integrals. . . . . . . . . . . 164 16.11 Error codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 16.12 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 16.13 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 166

17 Random Number Generation . . . . . . . . . . . . 167

17.1 General comments on random numbers . . . . . . . . . . . . . . . . . . . . 167 17.2 The Random Number Generator Interface. . . . . . . . . . . . . . . . . . 167 17.3 Random number generator initialization . . . . . . . . . . . . . . . . . . . 168 17.4 Sampling from a random number generator . . . . . . . . . . . . . . . . 168 17.5 Auxiliary random number generator functions . . . . . . . . . . . . . . 169 17.6 Random number environment variables . . . . . . . . . . . . . . . . . . . . 170 17.7 Copying random number generator state . . . . . . . . . . . . . . . . . . . 171 17.8 Reading and writing random number generator state. . . . . . . . 172 17.9 Random number generator algorithms . . . . . . . . . . . . . . . . . . . . . 172 17.10 Unix random number generators . . . . . . . . . . . . . . . . . . . . . . . . . 175 17.11 Other random number generators . . . . . . . . . . . . . . . . . . . . . . . . 176 17.12 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 17.13 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 17.14 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 182 17.15 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

18 Quasi-Random Sequences . . . . . . . . . . . . . . . 184

18.1 Quasi-random number generator initialization . . . . . . . . . . . . . . 184 18.2 Sampling from a quasi-random number generator . . . . . . . . . . . 184 18.3 Auxiliary quasi-random number generator functions. . . . . . . . . 184 18.4 Saving and resorting quasi-random number generator state . . 185 18.5 Quasi-random number generator algorithms . . . . . . . . . . . . . . . . 185 18.6 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 18.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

19 Random Number Distributions . . . . . . . . . . 187

19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 19.2 The Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 19.3 The Gaussian Tail Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 19.4 The Bivariate Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . 192 19.5 The Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 19.6 The Laplace Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 19.7 The Exponential Power Distribution . . . . . . . . . . . . . . . . . . . . . . . 195 19.8 The Cauchy Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 19.9 The Rayleigh Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 19.10 The Rayleigh Tail Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 19.11 The Landau Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 19.12 The Levy alpha-Stable Distributions. . . . . . . . . . . . . . . . . . . . . . 200 19.13 The Levy skew alpha-Stable Distribution . . . . . . . . . . . . . . . . . 201 19.14 The Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 19.15 The Flat (Uniform) Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 203 19.16 The Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 19.17 The Chi-squared Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 19.18 The F-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 19.19 The t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 19.20 The Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 19.21 The Logistic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 19.22 The Pareto Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 19.23 Spherical Vector Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 19.24 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 19.25 The Type-1 Gumbel Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 213 19.26 The Type-2 Gumbel Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 214 19.27 The Dirichlet Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 19.28 General Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 19.29 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 19.30 The Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 19.31 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 19.32 The Multinomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 19.33 The Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 222 19.34 The Pascal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 19.35 The Geometric Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 19.36 The Hypergeometric Distribution. . . . . . . . . . . . . . . . . . . . . . . . . 225 19.37 The Logarithmic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 19.38 Shuffling and Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 19.39 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 19.40 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 231

20 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

20.1 Mean, Standard Deviation and Variance . . . . . . . . . . . . . . . . . . . 232 20.2 Absolute deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 20.3 Higher moments (skewness and kurtosis) . . . . . . . . . . . . . . . . . . . 233 20.4 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 20.5 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 20.6 Weighted Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 20.7 Maximum and Minimum values . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 20.8 Median and Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 20.9 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 20.10 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 240

21 Histograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

21.1 The histogram struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 21.2 Histogram allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 21.3 Copying Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 21.4 Updating and accessing histogram elements . . . . . . . . . . . . . . . . 244 21.5 Searching histogram ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 21.6 Histogram Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 21.7 Histogram Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 21.8 Reading and writing histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 21.9 Resampling from histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 21.10 The histogram probability distribution struct. . . . . . . . . . . . . . 247 21.11 Example programs for histograms . . . . . . . . . . . . . . . . . . . . . . . . 249 21.12 Two dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 21.13 The 2D histogram struct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 21.14 2D Histogram allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 21.15 Copying 2D Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 21.16 Updating and accessing 2D histogram elements . . . . . . . . . . . . 252 21.17 Searching 2D histogram ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 21.18 2D Histogram Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 21.19 2D Histogram Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 21.20 Reading and writing 2D histograms. . . . . . . . . . . . . . . . . . . . . . . 255 21.21 Resampling from 2D histograms . . . . . . . . . . . . . . . . . . . . . . . . . . 256 21.22 Example programs for 2D histograms . . . . . . . . . . . . . . . . . . . . . 258

22 N-tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

22.1 The ntuple struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 22.2 Creating ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 22.3 Opening an existing ntuple file . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 22.4 Writing ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 22.5 Reading ntuples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 22.6 Closing an ntuple file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 22.7 Histogramming ntuple values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 22.8 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 22.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

23 Monte Carlo Integration . . . . . . . . . . . . . . . . 266

23.1 Interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 23.2 PLAIN Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 23.3 MISER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 23.4 VEGAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 23.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 23.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

24 Simulated Annealing. . . . . . . . . . . . . . . . . . . . 277

24.1 Simulated Annealing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 24.2 Simulated Annealing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 24.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

24.3.1 Trivial example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 24.3.2 Traveling Salesman Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 281

24.4 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

25 Ordinary Differential Equations . . . . . . . . . 285

25.1 Defining the ODE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 25.2 Stepping Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 25.3 Adaptive Step-size Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 25.4 Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 25.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 25.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

26 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 26.2 Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 26.3 Interpolation Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 26.4 Index Look-up and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 26.5 Evaluation of Interpolating Functions . . . . . . . . . . . . . . . . . . . . . . 296 26.6 Higher-level Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 26.7 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 26.8 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

27 Numerical Differentiation . . . . . . . . . . . . . . . 301

27.1 Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 27.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 27.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

28 Chebyshev Approximations . . . . . . . . . . . . . 304

28.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 28.2 Creation and Calculation of Chebyshev Series . . . . . . . . . . . . . . 304 28.3 Chebyshev Series Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 28.4 Derivatives and Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 28.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 28.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

29 Series Acceleration . . . . . . . . . . . . . . . . . . . . . 308

29.1 Acceleration functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 29.2 Acceleration functions without error estimation. . . . . . . . . . . . . 308 29.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 29.4 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

30 Wavelet Transforms. . . . . . . . . . . . . . . . . . . . . 312

30.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 30.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 30.3 Transform Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

30.3.1 Wavelet transforms in one dimension . . . . . . . . . . . . . . . . . . 313 30.3.2 Wavelet transforms in two dimension . . . . . . . . . . . . . . . . . . 314

30.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 30.5 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

31 Discrete Hankel Transforms . . . . . . . . . . . . . 319

31.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 31.2 Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 31.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

32 One dimensional Root-Finding . . . . . . . . . . 321

32.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 32.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 32.3 Initializing the Solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 32.4 Providing the function to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 32.5 Search Bounds and Guesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 32.6 Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 32.7 Search Stopping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 32.8 Root Bracketing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 32.9 Root Finding Algorithms using Derivatives . . . . . . . . . . . . . . . . . 328 32.10 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 32.11 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 333

33 One dimensional Minimization . . . . . . . . . . 334

33.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 33.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 33.3 Initializing the Minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 33.4 Providing the function to minimize . . . . . . . . . . . . . . . . . . . . . . . . 336 33.5 Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 33.6 Stopping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 33.7 Minimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 33.8 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 33.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

34 Multidimensional Root-Finding. . . . . . . . . . 341

34.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 34.2 Initializing the Solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 34.3 Providing the function to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 34.4 Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 34.5 Search Stopping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 34.6 Algorithms using Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 34.7 Algorithms without Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 34.8 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 34.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

35 Multidimensional Minimization. . . . . . . . . . 355

35.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 35.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 35.3 Initializing the Multidimensional Minimizer . . . . . . . . . . . . . . . . 356 35.4 Providing a function to minimize . . . . . . . . . . . . . . . . . . . . . . . . . . 357 35.5 Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 35.6 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 35.7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 35.8 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 35.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

36 Least-Squares Fitting . . . . . . . . . . . . . . . . . . . 366

36.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 36.2 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 36.3 Linear fitting without a constant term . . . . . . . . . . . . . . . . . . . . . 367 36.4 Multi-parameter fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 36.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 36.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

37 Nonlinear Least-Squares Fitting . . . . . . . . . 376

37.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 37.2 Initializing the Solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 37.3 Providing the Function to be Minimized . . . . . . . . . . . . . . . . . . . 377 37.4 Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 37.5 Search Stopping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 37.6 Minimization Algorithms using Derivatives . . . . . . . . . . . . . . . . . 380 37.7 Minimization Algorithms without Derivatives . . . . . . . . . . . . . . 381 37.8 Computing the covariance matrix of best fit parameters . . . . . 381 37.9 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 37.10 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 387

38 Physical Constants . . . . . . . . . . . . . . . . . . . . . 388

38.1 Fundamental Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 38.2 Astronomy and Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 38.3 Atomic and Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 38.4 Measurement of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 38.5 Imperial Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 38.6 Speed and Nautical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 38.7 Printers Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 38.8 Volume, Area and Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 38.9 Mass and Weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 38.10 Thermal Energy and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 38.11 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 38.12 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 38.13 Light and Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 38.14 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 38.15 Force and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 38.16 Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 38.17 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 38.18 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 396

39 IEEE floating-point arithmetic . . . . . . . . . . 397

39.1 Representation of floating point numbers. . . . . . . . . . . . . . . . . . . 397 39.2 Setting up your IEEE environment . . . . . . . . . . . . . . . . . . . . . . . . 399 39.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Appendix A Debugging Numerical Programs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

A.1 Using gdb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 A.2 Examining floating point registers. . . . . . . . . . . . . . . . . . . . . . . . . . 404 A.3 Handling floating point exceptions . . . . . . . . . . . . . . . . . . . . . . . . . 404 A.4 GCC warning options for numerical programs . . . . . . . . . . . . . . . 405 A.5 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Appendix B Contributors to GSL . . . . . . . . . . 407

Appendix C Autoconf Macros . . . . . . . . . . . . . 409

Appendix D GSL CBLAS Library. . . . . . . . . . 411

D.1 Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 D.2 Level 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 D.3 Level 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 D.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

Free Software Needs Free Documentation …. 424 GNU General Public License . . . . . . . . . . . . . . . . 426 GNU Free Documentation License . . . . . . . . . . . 430 Function Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Variable Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 Type Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Concept Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

1 Introduction

The GNU Scientific Library (GSL) is a collection of routines for numerical computing. The routines have been written from scratch in C, and present a modern Applications Programming Interface (API) for C programmers, allowing wrappers to be written for very high level languages. The source code is distributed under the GNU General Public License.

1.1 Routines available in GSL

The library covers a wide range of topics in numerical computing. Routines are available for the following areas,

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

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.

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 (see [GNU General Public License], page 426). 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/).

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.

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 (see [GNU General Public License], page 426).

1.5 Reporting Bugs

A list of known bugs can be found in the ‘BUGS’ file included in the GSL distribution. Details of compilation problems can be found in the ‘INSTALL’ file.

If you find a bug which is not listed in these files, please report it to bug-gsl@gnu.org. All bug reports should include:

• The version number of GSL

• The hardware and operating system

• The compiler used, including version number and compilation options

• A description of the bug behavior

• A short program which exercises the bug It is useful if you can check whether the same problem occurs when the library is compiled without optimization. Thank you.

Any errors or omissions in this manual can also be reported to the same address.

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/”.

1.7 Conventions used in this manual

This manual contains many examples which can be typed at the keyboard. A command entered at the terminal is shown like this,

     $ command

The first character on the line is the terminal prompt, and should not be typed. The dollar sign ‘$’ is used as the standard prompt in this manual, although some systems may use a different character.

The examples assume the use of the GNU operating system. There may be minor differences in the output on other systems. The commands for setting environment variables use the Bourne shell syntax of the standard GNU shell (bash).

2 Using the library

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

2.1 An Example Program

The following short program demonstrates the use of the library by computing the value of the Bessel function J_{0}(x) for x = 5,

     #include <stdio.h>
     #include <gsl/gsl_sf_bessel.h>
     int
     main (void)
     {
       double x = 5.0;
       double y = gsl_sf_bessel_J0 (x);
       printf ("J0(%g) = %.18e\n", x, y);
       return 0;
     }

The output is shown below, and should be correct to double-precision accuracy,

     J0(5) = -1.775967713143382920e-01

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

2.2 Compiling and Linking

The library header files are installed in their own ‘gsl’ directory. You should write any preprocessor include statements with a ‘gsl/’ directory prefix thus,

     #include <gsl/gsl_math.h>

If the directory is not installed on the standard search path of your compiler you will also need to provide its location to the preprocessor as a command line flag. The default location of the ‘gsl’ directory is ‘/usr/local/include/gsl’. A typical compilation command for a source file ‘example.c’ with the GNU C compiler gcc is,

     $ gcc -Wall -I/usr/local/include -c example.c

This results in an object file ‘example.o’. The default include path for gcc searches ‘/usr/local/include’ automatically so the -I option can actually be omitted when GSL is installed in its default location.

2.2.1 Linking programs with the library

The library is installed as a single file, ‘libgsl.a’. A shared version of the library ‘libgsl.so’ is also installed on systems that support shared libraries. The default location of these files is ‘/usr/local/lib’. If this directory is not on the standard search path of your linker you will also need to provide its location as a command line flag.

To link against the library you need to specify both the main library and a supporting cblas library, which provides standard basic linear algebra subroutines. A suitable cblas implementation is provided in the library ‘libgslcblas.a’ if your system does not provide one. The following example shows how to link an application with the library,

     $ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm

The default library path for gcc searches ‘/usr/local/lib’ automatically so the -L option can be omitted when GSL is installed in its default location.

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 Chapter 12 [BLAS Support], page 110.

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

2.4 ANSI C Compliance

The library is written in ANSI C and is intended to conform to the ANSI C standard (C89). It should be portable to any system with a working ANSI C compiler.

The library does not rely on any non-ANSI extensions in the interface it exports to the user. Programs you write using GSL can be ANSI compliant. Extensions which can be used in a way compatible with pure ANSI C are supported, however, via conditional compilation. This allows the library to take advantage of compiler extensions on those platforms which support them.

When an ANSI C feature is known to be broken on a particular system the library will exclude any related functions at compile-time. This should make it impossible to link a program that would use these functions and give incorrect results.

To avoid namespace conflicts all exported function names and variables have the prefix gsl_, while exported macros have the prefix GSL_.

2.5 Inline functions

The inline keyword is not part of the original ANSI C standard (C89) 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 Appendix C [Autoconf Macros], page 409.

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.

2.7 Portability functions

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

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

     /* Substitute gsl_hypot for missing system hypot */
     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

The application source files can then use the include command #include <config.h> to replace each occurrence of hypot by gsl_hypot when hypot is not available. This substitution can be made automatically if you use autoconf, see Appendix C [Autoconf Macros], page 409.

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

2.8 Alternative optimized functions

The main implementation of some functions in the library will not be optimal on all architectures. For example, there are several ways to compute a Gaussian random variate and their relative speeds are platform-dependent. In cases like this the library provides alternative implementations of these functions with the same interface. If you write your application using calls to the standard implementation you can select an alternative version later via a preprocessor definition. It is also possible to introduce your own optimized functions this way while retaining portability. The following lines demonstrate the use of a platform-dependent choice of methods for sampling from the Gaussian distribution,

     #ifdef SPARC
     #define gsl_ran_gaussian gsl_ran_gaussian_ratio_method
     #endif
     #ifdef INTEL
     #define gsl_ran_gaussian my_gaussian
     #endif

These lines would be placed in the configuration header file ‘config.h’ of the application, which should then be included by all the source files. Note that the alternative implementations will not produce bit-for-bit identical results, and in the case of random number distributions will produce an entirely different stream of random variates.

2.9 Support for different numeric types

Many functions in the library are defined for different numeric types. This feature is implemented by varying the name of the function with a type-related modifier—a primitive form of C++ templates. The modifier is inserted into the function name after the initial module prefix. The following table shows the function names defined for all the numeric types of an imaginary module gsl_foo with function fn,

gsl_foo_fn               double
gsl_foo_long_double_fn   long double
gsl_foo_float_fn         float
gsl_foo_long_fn          long
gsl_foo_ulong_fn         unsigned long
gsl_foo_int_fn           int
gsl_foo_uint_fn          unsigned int
gsl_foo_short_fn         short
gsl_foo_ushort_fn        unsigned short
gsl_foo_char_fn          char
gsl_foo_uchar_fn         unsigned char

The normal numeric precision double is considered the default and does not require a suffix. For example, the function gsl_stats_mean computes the mean of double precision numbers, while the function gsl_stats_int_mean computes the mean of integers.

A corresponding scheme is used for library defined types, such as gsl_vector and gsl_ matrix. In this case the modifier is appended to the type name. For example, if a module defines a new type-dependent struct or typedef gsl_foo it is modified for other types in the following way,

gsl_foo                  double
gsl_foo_long_double      long double
gsl_foo_float            float
gsl_foo_long             long
gsl_foo_ulong            unsigned long
gsl_foo_int              int
gsl_foo_uint             unsigned int
gsl_foo_short            short
gsl_foo_ushort           unsigned short
gsl_foo_char             char
gsl_foo_uchar            unsigned char

When a module contains type-dependent definitions the library provides individual header files for each type. The filenames are modified as shown in the below. For convenience the default header includes the definitions for all the types. To include only the double precision header file, or any other specific type, use its individual filename.

#include <gsl/gsl_foo.h>               All types
#include <gsl/gsl_foo_double.h>        double
#include <gsl/gsl_foo_long_double.h>   long double
#include <gsl/gsl_foo_float.h>         float
#include <gsl/gsl_foo_long.h>          long
#include <gsl/gsl_foo_ulong.h>         unsigned long

     #include <gsl/gsl_foo_int.h>           int
     #include <gsl/gsl_foo_uint.h>          unsigned int
     #include <gsl/gsl_foo_short.h>         short
     #include <gsl/gsl_foo_ushort.h>        unsigned short
     #include <gsl/gsl_foo_char.h>          char
     #include <gsl/gsl_foo_uchar.h>         unsigned char

2.10 Compatibility with C++

The library header files automatically define functions to have extern "C" linkage when included in C++ programs. This allows the functions to be called directly from C++.

To use C++ exception handling within user-defined functions passed to the library as parameters, the library must be built with the additional CFLAGS compilation option

‘-fexceptions’.

2.11 Aliasing of arrays

The library assumes that arrays, vectors and matrices passed as modifiable arguments are not aliased and do not overlap with each other. This removes the need for the library to handle overlapping memory regions as a special case, and allows additional optimizations to be used. If overlapping memory regions are passed as modifiable arguments then the results of such functions will be undefined. If the arguments will not be modified (for example, if a function prototype declares them as const arguments) then overlapping or aliased memory regions can be safely used.

2.12 Thread-safety

The library can be used in multi-threaded programs. All the functions are thread-safe, in the sense that they do not use static variables. Memory is always associated with objects and not with functions. For functions which use workspace objects as temporary storage the workspaces should be allocated on a per-thread basis. For functions which use table objects as read-only memory the tables can be used by multiple threads simultaneously. Table arguments are always declared const in function prototypes, to indicate that they may be safely accessed by different threads.

There are a small number of static global variables which are used to control the overall behavior of the library (e.g. whether to use range-checking, the function to call on fatal error, etc). These variables are set directly by the user, so they should be initialized once at program startup and not modified by different threads.

2.13 Deprecated Functions

From time to time, it may be necessary for the definitions of some functions to be altered or removed from the library. In these circumstances the functions will first be declared deprecated and then removed from subsequent versions of the library. Functions that are deprecated can be disabled in the current release by setting the preprocessor definition GSL_ DISABLE_DEPRECATED. This allows existing code to be tested for forwards compatibility.

2.14 Code Reuse

Where possible the routines in the library have been written to avoid dependencies between modules and files. This should make it possible to extract individual functions for use in your own applications, without needing to have the whole library installed. You may need to define certain macros such as GSL_ERROR and remove some #include statements in order to compile the files as standalone units. Reuse of the library code in this way is encouraged, subject to the terms of the GNU General Public License.

3 Error Handling

This chapter describes the way that GSL functions report and handle errors. By examining the status information returned by every function you can determine whether it succeeded or failed, and if it failed you can find out what the precise cause of failure was. You can also define your own error handling functions to modify the default behavior of the library.

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

3.1 Error Reporting

The library follows the thread-safe error reporting conventions of the posix Threads library. Functions return a non-zero error code to indicate an error and 0 to indicate success.

     int status = gsl_function (...)
     if (status) { /* an error occurred */
       .....
       /* status value specifies the type of error */
     }

The routines report an error whenever they cannot perform the task requested of them. For example, a root-finding function would return a non-zero error code if could not converge to the requested accuracy, or exceeded a limit on the number of iterations. Situations like this are a normal occurrence when using any mathematical library and you should check the return status of the functions that you call.

Whenever a routine reports an error the return value specifies the type of error. The return value is analogous to the value of the variable errno in the C library. The caller can examine the return code and decide what action to take, including ignoring the error if it is not considered serious.

In addition to reporting errors by return codes the library also has an error handler function gsl_error. This function is called by other library functions when they report an error, just before they return to the caller. The default behavior of the error handler is to print a message and abort the program,

     gsl: file.c:67: ERROR: invalid argument supplied by user
     Default GSL error handler invoked.
     Aborted

The purpose of the gsl_error handler is to provide a function where a breakpoint can be set that will catch library errors when running under the debugger. It is not intended for use in production programs, which should handle any errors using the return codes.

3.2 Error Codes

The error code numbers returned by library functions are defined in the file ‘gsl_errno.h’. They all have the prefix GSL_and expand to non-zero constant integer values. 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.

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’.

3.4 Using GSL error reporting in your own functions

If you are writing numerical functions in a program which also uses GSL code you may find it convenient to adopt the same error reporting conventions as in the library.

To report an error you need to call the function gsl_error with a string describing the error and then return an appropriate error code from gsl_errno.h, or a special value, such as NaN. For convenience the file ‘gsl_errno.h’ defines two macros which carry out these steps:

[Macro] GSL_ERROR (reason, gsl_errno)

This macro reports an error using the GSL conventions and returns a status value of gsl_errno. It expands to the following code fragment,

     gsl_error (reason, __FILE__, __LINE__, gsl_errno);
     return gsl_errno;

The macro definition in ‘gsl_errno.h’ actually wraps the code in a do { ... } while (0) block to prevent possible parsing problems.

Here is an example of how the macro could be used to report that a routine did not achieve a requested tolerance. To report the error the routine needs to return the error

code GSL_ETOL.
if (residual > tolerance)
  {
    GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
  }
[Macro] GSL_ERROR_VAL (reason, gsl_errno, value)

This macro is the same as GSL_ERROR but returns a user-defined value of value instead of an error code. It can be used for mathematical functions that return a floating point value.

The following example shows how to return a NaN at a mathematical singularity using

the GSL_ERROR_VAL macro,
if (x == 0)
  {
    GSL_ERROR_VAL("argument lies on singularity",
                  GSL_ERANGE, GSL_NAN);
  }

3.5 Examples

Here is an example of some code which checks the return value of a function where an error might be reported,

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
...
  int status;
  size_t n = 37;
  gsl_set_error_handler_off();
  status = gsl_fft_complex_radix2_forward (data, 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.

4 Mathematical Functions

This chapter describes basic mathematical functions. Some of these functions are present in system libraries, but the alternative versions given here can be used as a substitute when the system functions are not available.

The functions and macros described in this chapter are defined in the header file

‘gsl_math.h’.

4.1 Mathematical Constants

The library ensures that the standard bsd mathematical constants are defined. For reference, here is a list of the constants: M_E The base of exponentials, e M_LOG2E The base-2 logarithm of e, log_{2}(e) M_LOG10E The base-10 logarithm of e, log_{10}(e) M_SQRT2 The square root of two, \sqrt{2}

M_SQRT1_2

The square root of one-half, \sqrt{1/2}

M_SQRT3 The square root of three, \sqrt{3} M_PI The constant pi, π M_PI_2 Pi divided by two, π/2 M_PI_4 Pi divided by four, π/4 M_SQRTPI The square root of pi, \sqrt{π}

M_2_SQRTPI

Two divided by the square root of pi, 2/\sqrt{π}

M_1_PI The reciprocal of pi, 1/π M_2_PI Twice the reciprocal of pi, 2/π 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(π) M_EULER Euler’s constant, γ

4.2 Infinities and Not-a-number

[Macro] GSL_POSINF

This macro contains the IEEE representation of positive infinity, +∞. It is computed from the expression +1.0/0.0.

[Macro] GSL_NEGINF

This macro contains the IEEE representation of negative infinity, −∞. 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.

4.3 Elementary Functions

The following routines provide portable implementations of functions found in the BSD math library. When native versions are not available the functions described here can be used instead. The substitution can be made automatically if you use autoconf to compile your application (see Section 2.7 [Portability functions], page 7).

[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).

4.4 Small integer powers

A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides a simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow or underflow conditions.

[Function] double gsl_pow_int (double x, int n)

This routine computes the power x^{n} for integer n. The power is computed efficiently— for example, x^{8} is computed as ((x^{2})^{2})^{2}, requiring only 3 multiplications. A version of this function which also computes the numerical error in the result is available as

gsl_sf_pow_int_e.

[Function] double gsl_pow_2 (const double x)

[Function] double gsl_pow_3 (const double x)

[Function] double gsl_pow_4 (const double x)

[Function] double gsl_pow_5 (const double x)

[Function] double gsl_pow_6 (const double x)

[Function] double gsl_pow_7 (const double x)

[Function] double gsl_pow_8 (const double x)

[Function] double gsl_pow_9 (const double x)

These functions can be used to compute small integer powers x^{2}, x^{3}, etc. efficiently. The functions will be inlined when possible so that use of these functions should be as efficient as explicitly writing the corresponding product expression.

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

4.5 Testing the Sign of Numbers

[Macro] GSL_SIGN (x)

This macro returns the sign of x. It is defined as ((x) >= 0 ? 1 : -1). Note that with this definition the sign of zero is positive (regardless of its ieee sign bit).

4.6 Testing for Odd and Even Numbers

[Macro] GSL_IS_ODD (n)

This macro evaluates to 1 if n is odd and 0 if n is even. The argument n must be of integer type.

[Macro] GSL_IS_EVEN (n)

This macro is the opposite of GSL_IS_ODD(n). It evaluates to 1 if n is even and 0 if n is odd. The argument n must be of integer type.

4.7 Maximum and Minimum functions

[Macro] GSL_MAX (a, b)

This macro returns the maximum of a and b. It is defined as ((a) > (b) ? (a):(b)).

[Macro] GSL_MIN (a, b)

This macro returns the minimum of a and b. It is defined as ((a) < (b) ? (a):(b)).

[Function] extern inline double GSL_MAX_DBL (double a, double b)

This function returns the maximum of the double precision numbers a and b using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro GSL_MAX will be automatically substituted.

[Function] extern inline double GSL_MIN_DBL (double a, double b)

This function returns the minimum of the double precision numbers a and b using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro GSL_MIN will be automatically substituted.

[Function] extern inline int GSL_MAX_INT (int a, int b)
[Function] extern inline int GSL_MIN_INT (int a, int b)

These functions return the maximum or minimum of the integers a and b using an inline function. On platforms where inline functions are not available the macros GSL_MAX or GSL_MIN will be automatically substituted.

[Function] extern inline long double GSL_MAX_LDBL (long double a, long

double b)

[Function] extern inline long double GSL_MIN_LDBL (long double a, long

double b) These functions return the maximum or minimum of the long doubles a and b using an inline function. On platforms where inline functions are not available the macros GSL_MAX or GSL_MIN will be automatically substituted.

4.8 Approximate Comparison of Floating Point Numbers

It is sometimes useful to be able to compare two floating point numbers approximately, to allow for rounding and truncation errors. The following function implements the approximate floating-point comparison algorithm proposed by D.E. Knuth in Section 4.2.2 of Seminumerical Algorithms (3rd edition).

[Function] int gsl_fcmp (double x, double y, double epsilon)

This function determines whether x and y are approximately equal to a relative accuracy epsilon. The relative accuracy is measured using an interval of size 2δ, where δ = 2^{k}ϵ 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.

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’.

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 + iy.

[Function] gsl_complex gsl_complex_polar (double r, double theta)

This function returns the complex number z = r exp(iθ) = r(cos(θ) + i sin(θ)) 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.

1 Note that the first edition uses different definitions.

[Macro] GSL_SET_REAL (zp,x)
[Macro] GSL_SET_IMAG (zp,y)

These macros allow the real and imaginary parts of the complex number pointed to by zp to be set independently.

5.2 Properties of complex numbers

[Function] double gsl_complex_arg (gsl complex z)

This function returns the argument of the complex number z, arg(z), where −π < arg(z) ≤ π.

[Function] double gsl_complex_abs (gsl complex z)

This function returns the magnitude of the complex number z, |z|.

[Function] double gsl_complex_abs2 (gsl complex z)

This function returns the squared magnitude of the complex number z, |z|^{2}.

[Function] double gsl_complex_logabs (gsl complex z)

This function returns the natural logarithm of the magnitude of the complex number z, log |z|. It allows an accurate evaluation of log |z| when |z| is close to one. The direct evaluation of log(gsl_complex_abs(z)) would lead to a loss of precision in this case.

5.3 Complex arithmetic operators

[Function] gsl_complex gsl_complex_add (gsl complex a, gsl complex b)

This function returns the sum of the complex numbers a and b, z = a + b.

[Function] gsl_complex gsl_complex_sub (gsl complex a, gsl complex b)

This function returns the difference of the complex numbers a and b, z = a − b.

[Function] gsl_complex gsl_complex_mul (gsl complex a, gsl complex b)

This function returns the product of the complex numbers a and b, z = ab.

[Function] gsl_complex gsl_complex_div (gsl complex a, gsl complex b)

This function returns the quotient of the complex numbers a and b, z = a/b.

[Function] gsl_complex gsl_complex_add_real (gsl complex a, double x)

This function returns the sum of the complex number a and the real number x, z = a + x.

[Function] gsl_complex gsl_complex_sub_real (gsl complex a, double x)

This function returns the difference of the complex number a and the real number x, z = a − x.

[Function] gsl_complex gsl_complex_mul_real (gsl complex a, double x)

This function returns the product of the complex number a and the real number x, z = ax.

[Function] gsl_complex gsl_complex_div_real (gsl complex a, double x)

This function returns the quotient of the complex number a and the real number x, z = a/x.

[Function] gsl_complex gsl_complex_add_imag (gsl complex a, double y)

This function returns the sum of the complex number a and the imaginary number iy, z = a + iy.

[Function] gsl_complex gsl_complex_sub_imag (gsl complex a, double y)

This function returns the difference of the complex number a and the imaginary number iy, z = a − iy.

[Function] gsl_complex gsl_complex_mul_imag (gsl complex a, double y)

This function returns the product of the complex number a and the imaginary number iy, z = a ∗ (iy).

[Function] gsl_complex gsl_complex_div_imag (gsl complex a, double y)

This function returns the quotient of the complex number a and the imaginary number iy, z = a/(iy).

[Function] gsl_complex gsl_complex_conjugate (gsl complex z)

This function returns the complex conjugate of the complex number z, z^{∗} = x − iy.

[Function] gsl_complex gsl_complex_inverse (gsl complex z)

This function returns the inverse, or reciprocal, of the complex number z, 1/z = (x − iy)/(x^{2} + y^{2}).

[Function] gsl_complex gsl_complex_negative (gsl complex z)

This function returns the negative of the complex number z, −z = (−x) + i(−y).

5.4 Elementary Complex Functions

[Function] gsl_complex gsl_complex_sqrt (gsl complex z)

This function returns the square root of the complex number z, \sqrt{z}. The branch cut is the negative real axis. The result always lies in the right half of the complex plane.

[Function] gsl_complex gsl_complex_sqrt_real (double x)

This function returns the complex square root of the real number x, where x may be negative.

[Function] gsl_complex gsl_complex_pow (gsl complex z, gsl complex a)

The function returns the complex number z raised to the complex power a, z^{a}. This is computed as exp(log(z) ∗ a) using complex logarithms and complex exponentials.

[Function] gsl_complex gsl_complex_pow_real (gsl complex z, double x)

This function returns the complex number z raised to the real power x, z^{x}.

[Function] gsl_complex gsl_complex_exp (gsl complex z)

This function returns the complex exponential of the complex number z, exp(z).

[Function] gsl_complex gsl_complex_log (gsl complex z)

This function returns the complex natural logarithm (base e) of the complex number z, log(z). The branch cut is the negative real axis.

[Function] gsl_complex gsl_complex_log10 (gsl complex z)

This function returns the complex base-10 logarithm of the complex number z, log_{10}(z).

[Function] gsl_complex gsl_complex_log_b (gsl complex z, gsl complex b)

This function returns the complex base-b logarithm of the complex number z, log_{b}(z). This quantity is computed as the ratio log(z)/ log(b).

5.5 Complex Trigonometric Functions

[Function] gsl_complex gsl_complex_sin (gsl complex z)

This function returns the complex sine of the complex number z, sin(z) = (exp(iz) − exp(−iz))/(2i).

[Function] gsl_complex gsl_complex_cos (gsl complex z)

This function returns the complex cosine of the complex number z, cos(z) = (exp(iz)+ exp(−iz))/2.

[Function] gsl_complex gsl_complex_tan (gsl complex z)

This function returns the complex tangent of the complex number z, tan(z) = sin(z)/ cos(z).

[Function] gsl_complex gsl_complex_sec (gsl complex z)

This function returns the complex secant of the complex number z, sec(z) = 1/ cos(z).

[Function] gsl_complex gsl_complex_csc (gsl complex z)

This function returns the complex cosecant of the complex number z, csc(z) = 1/ sin(z).

[Function] gsl_complex gsl_complex_cot (gsl complex z)

This function returns the complex cotangent of the complex number z, cot(z) = 1/ tan(z).

5.6 Inverse Complex Trigonometric Functions

[Function] gsl_complex gsl_complex_arcsin (gsl complex z)

This function returns the complex arcsine of the complex number z, arcsin(z). The branch cuts are on the real axis, less than −1 and greater than 1.

[Function] gsl_complex gsl_complex_arcsin_real (double z)

This function returns the complex arcsine of the real number z, arcsin(z). For z between −1 and 1, the function returns a real value in the range [−π/2, π/2]. For z less than −1 the result has a real part of −π/2 and a positive imaginary part. For z greater than 1 the result has a real part of π/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, π]. For z less than −1 the result has a real part of π and a negative imaginary part. For z greater than 1 the result is purely imaginary and positive.

[Function] gsl_complex gsl_complex_arctan (gsl complex z)

This function returns the complex arctangent of the complex number z, arctan(z). The branch cuts are on the imaginary axis, below −i and above i.

[Function] gsl_complex gsl_complex_arcsec (gsl complex z)

This function returns the complex arcsecant of the complex number z, arcsec(z) = arccos(1/z).

[Function] gsl_complex gsl_complex_arcsec_real (double z)

This function returns the complex arcsecant of the real number z, arcsec(z) = arccos(1/z).

[Function] gsl_complex gsl_complex_arccsc (gsl complex z)

This function returns the complex arccosecant of the complex number z, arccsc(z) = arcsin(1/z).

[Function] gsl_complex gsl_complex_arccsc_real (double z)

This function returns the complex arccosecant of the real number z, arccsc(z) = arcsin(1/z).

[Function] gsl_complex gsl_complex_arccot (gsl complex z)

This function returns the complex arccotangent of the complex number z, arccot(z) = arctan(1/z).

5.7 Complex Hyperbolic Functions

[Function] gsl_complex gsl_complex_sinh (gsl complex z)

This function returns the complex hyperbolic sine of the complex number z, sinh(z) = (exp(z) − exp(−z))/2.

[Function] gsl_complex gsl_complex_cosh (gsl complex z)

This function returns the complex hyperbolic cosine of the complex number z, cosh(z) = (exp(z) + exp(−z))/2.

[Function] gsl_complex gsl_complex_tanh (gsl complex z)

This function returns the complex hyperbolic tangent of the complex number z, tanh(z) = sinh(z)/ cosh(z).

[Function] gsl_complex gsl_complex_sech (gsl complex z)

This function returns the complex hyperbolic secant of the complex number z, sech(z) = 1/ cosh(z).

[Function] gsl_complex gsl_complex_csch (gsl complex z)

This function returns the complex hyperbolic cosecant of the complex number z, csch(z) = 1/ sinh(z).

[Function] gsl_complex gsl_complex_coth (gsl complex z)

This function returns the complex hyperbolic cotangent of the complex number z, coth(z) = 1/ tanh(z).

5.8 Inverse Complex Hyperbolic Functions

[Function] gsl_complex gsl_complex_arcsinh (gsl complex z)

This function returns the complex hyperbolic arcsine of the complex number z, arcsinh(z). The branch cuts are on the imaginary axis, below −i and above i.

[Function] gsl_complex gsl_complex_arccosh (gsl complex z)

This function returns the complex hyperbolic arccosine of the complex number z, arccosh(z). The branch cut is on the real axis, less than 1.

[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).

5.9 References and Further Reading

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

T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang, “Implementing Complex Elementary Functions Using Exception Handling”, ACM Transactions on Mathematical Software, Volume 20 (1994), pp 215–244, Corrigenda, p553 T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang, “Implementing the complex arcsin and arccosine functions using exception handling”, ACM Transactions on Mathematical Software, Volume 23 (1997) pp 299–335

The general formulas and details of branch cuts can be found in the following books, Abramowitz and Stegun, Handbook of Mathematical Functions, “Circular Functions in Terms of Real and Imaginary Parts”, Formulas 4.3.55–58, “Inverse Circular Functions in Terms of Real and Imaginary Parts”, Formulas 4.4.37–39, “Hyperbolic Functions in Terms of Real and Imaginary Parts”, Formulas 4.5.49–52, “Inverse Hyperbolic Functions—relation to Inverse Circular Functions”, Formulas 4.6.14–19. Dave Gillespie, Calc Manual, Free Software Foundation, ISBN 1-882114-18-3

6 Polynomials

This chapter describes functions for evaluating and solving polynomials. There are routines for finding real and complex roots of quadratic and cubic equations using analytic methods. An iterative polynomial solver is also available for finding the roots of general polynomials with real coefficients (of any order). The functions are declared in the header file gsl_

poly.h.

6.1 Polynomial Evaluation

[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} + … + c[len − 1]x^{len−1} using Horner’s method for stability. The function is inlined when possible.

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.

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,

ax^{2} + bx + 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} −4ac. 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,

az^{2} + bz + c = 0

The number of complex roots is returned (either one or two) and the locations of the roots are stored in z0 and z1. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components. If only one real root is found (i.e. if a = 0) then it is stored in z0.

6.4 Cubic Equations

[Function] int gsl_poly_solve_cubic (double a, double b, double c, double * x0,

double * x1, double * x2) This function finds the real roots of the cubic equation,

x^{3} + ax^{2} + bx + 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} + az^{2} + bz + c = 0

The number of complex roots is returned (always three) and the locations of the roots are stored in z0, z1 and z2. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components.

6.5 General Polynomial Equations

The roots of polynomial equations cannot be found analytically beyond the special cases of the quadratic, cubic and quartic equation. The algorithm described in this section uses an iterative method to find the approximate locations of roots of higher order polynomials.

[Function] gsl_poly_complex_workspace *
   gsl_poly_complex_workspace_alloc (size t n)

This function allocates space for a gsl_poly_complex_workspace struct and a workspace suitable for solving a polynomial with n coefficients using the routine

gsl_poly_complex_solve.

The function returns a pointer to the newly allocated gsl_poly_complex_workspace if no errors were detected, and a null pointer in the case of error.

[Function] void gsl_poly_complex_workspace_free

(gsl poly complex workspace * w) This function frees all the memory associated with the workspace w.

[Function] int gsl_poly_complex_solve (const double * a, size t n,

gsl poly complex workspace * w, gsl complex packed ptr z) This function computes the roots of the general polynomial P (x) = a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{n−1}x^{n−1} using balanced-QR reduction of the companion matrix. The parameter n specifies the length of the coefficient array. The coefficient of the highest order term must be non-zero. The function requires a workspace w of the appropriate size. The n − 1 roots are returned in the packed complex array z of length 2(n − 1), alternating real and imaginary parts. The function returns GSL_SUCCESS if all the roots are found and GSL_EFAILED if the QR reduction does not converge. Note that due to finite precision, roots of higher multiplicity are returned as a cluster of simple roots with reduced accuracy. The solution of polynomials with higher-order roots requires specialized algorithms that take the multiplicity structure into account (see e.g. Z. Zeng, Algorithm 835, ACM Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp 218–236).

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^{2πi/5}, e^{4πi/5}, e^{6πi/5}, e^{8πi/5}

The following program will find these roots.

#include <stdio.h>
#include <gsl/gsl_poly.h>
int
main (void)
{
  int i;
  /* coefficients of P(x) =  -1 + x^5  */
  double a[6] = { -1, 0, 0, 0, 0, 1 };
  double z[10];
  gsl_poly_complex_workspace * w
      = gsl_poly_complex_workspace_alloc (6);

   gsl_poly_complex_solve (a, 6, w, z);
   gsl_poly_complex_workspace_free (w);
   for (i = 0; i < 5; i++)
     {
       printf ("z%d = %+.18f %+.18f\n",
               i, z[2*i], z[2*i+1]);
     }
   return 0;
 }

The output of the program is,

 $ ./a.out
 z0 = -0.809016994374947451 +0.587785252292473137
 z1 = -0.809016994374947451 -0.587785252292473137
 z2 = +0.309016994374947451 +0.951056516295153642
 z3 = +0.309016994374947451 -0.951056516295153642
 z4 = +1.000000000000000000 +0.000000000000000000

which agrees with the analytic result, z_{n} = exp(2πni/5).

6.7 References and Further Reading

The balanced-QR method and its error analysis are described in the following papers, R.S. Martin, G. Peters and J.H. Wilkinson, “The QR Algorithm for Real Hessenberg Matrices”, Numerische Mathematik, 14 (1970), 219–231. B.N. Parlett and C. Reinsch, “Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors”, Numerische Mathematik, 13 (1969), 293–304. A. Edelman and H. Murakami, “Polynomial roots from companion matrix eigenvalues”, Mathematics of Computation, Vol. 64, No. 210 (1995), 763–776.

The formulas for divided differences are given in Abramowitz and Stegun, Abramowitz and Stegun, Handbook of Mathematical Functions, Sections 25.1.4 and 25.2.26.

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’.

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 errorhandling functions return GSL_SUCCESS.

7.2 The gsl sf result struct

The error handling form of the special functions always calculate an error estimate along with the value of the result. Therefore, structures are provided for amalgamating a value and error estimate. These structures are declared in the header file ‘gsl_sf_result.h’.

The 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;

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 1 × 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.

7.4 Airy Functions and Derivatives

The Airy functions Ai(x) and Bi(x) are defined by the integral representations,

\begin{aligned}Ai(x) & = \frac{1}{π} ∫_{0}^{∞} cos(t^{3}/3 + xt) dt, \\ Bi(x) & = \frac{1}{π} ∫_{0}^{∞}(e^{−t^{3}/3} + sin(t^{3}/3 + xt)) dt. \end{aligned}

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

7.4.1 Airy Functions

[Function] double gsl_sf_airy_Ai (double x, gsl mode t mode)

[Function] int gsl_sf_airy_Ai_e (double x, gsl mode t mode, gsl sf result *

        result)

These routines compute the Airy function Ai(x) with an accuracy specified by mode.

[Function] double gsl_sf_airy_Bi (double x, gsl mode t mode)

[Function] int gsl_sf_airy_Bi_e (double x, gsl mode t mode, gsl sf result *

   result)

These routines compute the Airy function Bi(x) with an accuracy specified by mode.

[Function] double gsl_sf_airy_Ai_scaled (double x, gsl mode t mode)
[Function] int gsl_sf_airy_Ai_scaled_e (double x, gsl mode t mode,

gsl sf result * result) These routines compute a scaled version of the Airy function S_{A}(x)Ai(x). For x > 0 the scaling factor S_{A}(x) is exp(+(2/3)x^{3/2}), and is 1 for x < 0.

[Function] double gsl_sf_airy_Bi_scaled (double x, gsl mode t mode)
[Function] int gsl_sf_airy_Bi_scaled_e (double x, gsl mode t mode,

gsl sf result * result) These routines compute a scaled version of the Airy function S_{B}(x)Bi(x). For x > 0 the scaling factor S_{B}(x) is exp(−(2/3)x^{3/2}), and is 1 for x < 0.

7.4.2 Derivatives of Airy Functions

[Function] double gsl_sf_airy_Ai_deriv (double x, gsl mode t mode)

[Function] int gsl_sf_airy_Ai_deriv_e (double x, gsl mode t mode, gsl sf result

   * result)

These routines compute the Airy function derivative Ai′(x) with an accuracy specified by mode.

[Function] double gsl_sf_airy_Bi_deriv (double x, gsl mode t mode)

[Function] int gsl_sf_airy_Bi_deriv_e (double x, gsl mode t mode, gsl sf result

   * result)

These routines compute the Airy function derivative Bi′(x) with an accuracy specified by mode.

[Function] double gsl_sf_airy_Ai_deriv_scaled (double x, gsl mode t mode)
[Function] int gsl_sf_airy_Ai_deriv_scaled_e (double x, gsl mode t mode,

gsl sf result * result) These routines compute the scaled Airy function derivative S_{A}(x)Ai′(x). For x > 0 the scaling factor S_{A}(x) is exp(+(2/3)x^{3/2}), and is 1 for x < 0.

[Function] double gsl_sf_airy_Bi_deriv_scaled (double x, gsl mode t mode)
[Function] int gsl_sf_airy_Bi_deriv_scaled_e (double x, gsl mode t mode,

gsl sf result * result) These routines compute the scaled Airy function derivative S_{B}(x)Bi′(x). For x > 0 the scaling factor S_{B}(x) is exp(−(2/3)x^{3/2}), and is 1 for x < 0.

7.4.3 Zeros of Airy Functions

[Function] double gsl_sf_airy_zero_Ai (unsigned int s)

[Function] int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl sf result * result)

These routines compute the location of the s-th zero of the Airy function Ai(x).

[Function] double gsl_sf_airy_zero_Bi (unsigned int s)

[Function] int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl sf result * result)

These routines compute the location of the s-th zero of the Airy function Bi(x).

7.4.4 Zeros of Derivatives of Airy Functions

[Function] double gsl_sf_airy_zero_Ai_deriv (unsigned int s)

[Function] int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s, gsl sf result *

   result)

These routines compute the location of the s-th zero of the Airy function derivative Ai′(x).

[Function] double gsl_sf_airy_zero_Bi_deriv (unsigned int s)

[Function] int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s, gsl sf result *

   result)

These routines compute the location of the s-th zero of the Airy function derivative Bi′(x).

7.5 Bessel Functions

The routines described in this section compute the Cylindrical Bessel functions J_{n}(x), Y_{n}(x), Modified cylindrical Bessel functions I_{n}(x), K_{n}(x), Spherical Bessel functions j_{l}(x), y_{l}(x), and Modified Spherical Bessel functions i_{l}(x), k_{l}(x). For more information see Abramowitz & Stegun, Chapters 9 and 10. The Bessel functions are defined in the header

file ‘gsl_sf_bessel.h’.

7.5.1 Regular Cylindrical Bessel Functions

[Function] double gsl_sf_bessel_J0 (double x)

[Function] int gsl_sf_bessel_J0_e (double x, gsl sf result * result)

These routines compute the regular cylindrical Bessel function of zeroth order, J_{0}(x).

[Function] double gsl_sf_bessel_J1 (double x)

[Function] int gsl_sf_bessel_J1_e (double x, gsl sf result * result)

These routines compute the regular cylindrical Bessel function of first order, J_{1}(x).

[Function] double gsl_sf_bessel_Jn (int n, double x)

[Function] int gsl_sf_bessel_Jn_e (int n, double x, gsl sf result * result)

These routines compute the regular cylindrical Bessel function of order n, J_{n}(x).

[Function] int gsl_sf_bessel_Jn_array (int nmin, int nmax, double x, double
   result_array[])

This routine computes the values of the regular cylindrical Bessel functions J_{n}(x) for n from nmin to nmax inclusive, storing the results in the array result array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

7.5.2 Irregular Cylindrical Bessel Functions

[Function] double gsl_sf_bessel_Y0 (double x)

[Function] int gsl_sf_bessel_Y0_e (double x, gsl sf result * result)

These routines compute the irregular cylindrical Bessel function of zeroth order, Y_{0}(x), for x > 0.

[Function] double gsl_sf_bessel_Y1 (double x)

[Function] int gsl_sf_bessel_Y1_e (double x, gsl sf result * result)

These routines compute the irregular cylindrical Bessel function of first order, Y_{1}(x), for x > 0.

[Function] double gsl_sf_bessel_Yn (int n,double x)

[Function] int gsl_sf_bessel_Yn_e (int n,double x, gsl sf result * result)

These routines compute the irregular cylindrical Bessel function of order n, Y_{n}(x), for x > 0.

[Function] int gsl_sf_bessel_Yn_array (int nmin, int nmax, double x, double
   result_array[])

This routine computes the values of the irregular cylindrical Bessel functions Y_{n}(x) for n from nmin to nmax inclusive, storing the results in the array result array. The domain of the function is x > 0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

7.5.3 Regular Modified Cylindrical Bessel Functions

[Function] double gsl_sf_bessel_I0 (double x)

[Function] int gsl_sf_bessel_I0_e (double x, gsl sf result * result)

These routines compute the regular modified cylindrical Bessel function of zeroth order, I_{0}(x).

[Function] double gsl_sf_bessel_I1 (double x)

[Function] int gsl_sf_bessel_I1_e (double x, gsl sf result * result)

These routines compute the regular modified cylindrical Bessel function of first order, I_{1}(x).

[Function] double gsl_sf_bessel_In (int n, double x)

[Function] int gsl_sf_bessel_In_e (int n, double x, gsl sf result * result)

These routines compute the regular modified cylindrical Bessel function of order n, I_{n}(x).

[Function] int gsl_sf_bessel_In_array (int nmin, int nmax, double x, double
   result_array[])

This routine computes the values of the regular modified cylindrical Bessel functions I_{n}(x) for n from nmin to nmax inclusive, storing the results in the array result array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

[Function] double gsl_sf_bessel_I0_scaled (double x)
[Function] int gsl_sf_bessel_I0_scaled_e (double x, gsl sf result * result)

These routines compute the scaled regular modified cylindrical Bessel function of zeroth order exp(−|x|)I_{0}(x).

[Function] double gsl_sf_bessel_I1_scaled (double x)
[Function] int gsl_sf_bessel_I1_scaled_e (double x, gsl sf result * result)

These routines compute the scaled regular modified cylindrical Bessel function of first order exp(−|x|)I_{1}(x).

[Function] double gsl_sf_bessel_In_scaled (int n, double x)

[Function] int gsl_sf_bessel_In_scaled_e (int n, double x, gsl sf result *

   result)

These routines compute the scaled regular modified cylindrical Bessel function of order n, exp(−|x|)I_{n}(x)

[Function] int gsl_sf_bessel_In_scaled_array (int nmin, int nmax, double x,
   double result_array[])

This routine computes the values of the scaled regular cylindrical Bessel functions exp(−|x|)I_{n}(x) for n from nmin to nmax inclusive, storing the results in the array result array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

7.5.4 Irregular Modified Cylindrical Bessel Functions

[Function] double gsl_sf_bessel_K0 (double x)

[Function] int gsl_sf_bessel_K0_e (double x, gsl sf result * result)

These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_{0}(x), for x > 0.

[Function] double gsl_sf_bessel_K1 (double x)

[Function] int gsl_sf_bessel_K1_e (double x, gsl sf result * result)

These routines compute the irregular modified cylindrical Bessel function of first order, K_{1}(x), for x > 0.

[Function] double gsl_sf_bessel_Kn (int n, double x)

[Function] int gsl_sf_bessel_Kn_e (int n, double x, gsl sf result * result)

These routines compute the irregular modified cylindrical Bessel function of order n, K_{n}(x), for x > 0.

[Function] int gsl_sf_bessel_Kn_array (int nmin, int nmax, double x, double
   result_array[])

This routine computes the values of the irregular modified cylindrical Bessel functions K_{n}(x) for n from nmin to nmax inclusive, storing the results in the array result array. The start of the range nmin must be positive or zero. The domain of the function is x > 0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

[Function] double gsl_sf_bessel_K0_scaled (double x)
[Function] int gsl_sf_bessel_K0_scaled_e (double x, gsl sf result * result)

These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order exp(x)K_{0}(x) for x > 0.

[Function] double gsl_sf_bessel_K1_scaled (double x)
[Function] int gsl_sf_bessel_K1_scaled_e (double x, gsl sf result * result)

These routines compute the scaled irregular modified cylindrical Bessel function of first order exp(x)K_{1}(x) for x > 0.

[Function] double gsl_sf_bessel_Kn_scaled (int n, double x)

[Function] int gsl_sf_bessel_Kn_scaled_e (int n, double x, gsl sf result *

   result)

These routines compute the scaled irregular modified cylindrical Bessel function of order n, exp(x)K_{n}(x), for x > 0.

[Function] int gsl_sf_bessel_Kn_scaled_array (int nmin, int nmax, double x,
   double result_array[])

This routine computes the values of the scaled irregular cylindrical Bessel functions exp(x)K_{n}(x) for n from nmin to nmax inclusive, storing the results in the array result array. The start of the range nmin must be positive or zero. The domain of the function is x > 0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

7.5.5 Regular Spherical Bessel Functions

[Function] double gsl_sf_bessel_j0 (double x)

[Function] int gsl_sf_bessel_j0_e (double x, gsl sf result * result)

These routines compute the regular spherical Bessel function of zeroth order, j_{0}(x) = sin(x)/x.

[Function] double gsl_sf_bessel_j1 (double x)

[Function] int gsl_sf_bessel_j1_e (double x, gsl sf result * result)

These routines compute the regular spherical Bessel function of first order, j_{1}(x) = (sin(x)/x − cos(x))/x.

[Function] double gsl_sf_bessel_j2 (double x)

[Function] int gsl_sf_bessel_j2_e (double x, gsl sf result * result)

These routines compute the regular spherical Bessel function of second order, j_{2}(x) = ((3/x^{2} − 1) sin(x) − 3 cos(x)/x)/x.

[Function] double gsl_sf_bessel_jl (int l, double x)

[Function] int gsl_sf_bessel_jl_e (int l, double x, gsl sf result * result)

These routines compute the regular spherical Bessel function of order l, j_{l}(x), for l ≥ 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.

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.

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{π/(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.

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{π/(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.

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 ν, J_{ν}(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 ν, J_{ν}(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 overwritten with the values of J_{ν}(x_{i}).

7.5.10 Irregular Bessel Functions—Fractional Order

[Function] double gsl_sf_bessel_Ynu (double nu, double x)

[Function] int gsl_sf_bessel_Ynu_e (double nu, double x, gsl sf result *

   result)

These routines compute the irregular cylindrical Bessel function of fractional order ν, Y_{ν}(x).

7.5.11 Regular Modified Bessel Functions—Fractional Order

[Function] double gsl_sf_bessel_Inu (double nu, double x)

[Function] int gsl_sf_bessel_Inu_e (double nu, double x, gsl sf result *

   result)

These routines compute the regular modified Bessel function of fractional order ν, I_{ν}(x) for x > 0, ν > 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 ν, exp(−|x|)I_{ν}(x) for x > 0, ν > 0.

7.5.12 Irregular Modified Bessel Functions—Fractional Order

[Function] double gsl_sf_bessel_Knu (double nu, double x)

[Function] int gsl_sf_bessel_Knu_e (double nu, double x, gsl sf result *

   result)

These routines compute the irregular modified Bessel function of fractional order ν, K_{ν}(x) for x > 0, ν > 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 ν, ln(K_{ν}(x)) for x > 0, ν > 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 ν, exp(+|x|)K_{ν}(x) for x > 0, ν > 0.

7.5.13 Zeros of Regular Bessel Functions

[Function] double gsl_sf_bessel_zero_J0 (unsigned int s)

[Function] int gsl_sf_bessel_zero_J0_e (unsigned int s, gsl sf result * result)

These routines compute the location of the s-th positive zero of the Bessel function J_{0}(x).

[Function] double gsl_sf_bessel_zero_J1 (unsigned int s)

[Function] int gsl_sf_bessel_zero_J1_e (unsigned int s, gsl sf result * result)

These routines compute the location of the s-th positive zero of the Bessel function J_{1}(x).

[Function] double gsl_sf_bessel_zero_Jnu (double nu, unsigned int s)
[Function] int gsl_sf_bessel_zero_Jnu_e (double nu, unsigned int s,

gsl sf result * result) These routines compute the location of the s-th positive zero of the Bessel function J_{ν}(x). The current implementation does not support negative values of nu.

7.6 Clausen Functions

The Clausen function is defined by the following integral,

Cl_{2}(x) = − ∫_{0}^{x} dt log(2 sin(t/2))

It is related to the dilogarithm by Cl_{2}(θ) = Im(Li_{2}(e^{iθ})). The Clausen functions are declared in the header file ‘gsl_sf_clausen.h’.

[Function] double gsl_sf_clausen (double x)

[Function] int gsl_sf_clausen_e (double x, gsl sf result * result)

These routines compute the Clausen integral Cl_{2}(x).

7.7 Coulomb Functions

The prototypes of the Coulomb functions are declared in the header file ‘gsl_sf_coulomb.h’. Both bound state and scattering solutions are available.

7.7.1 Normalized Hydrogenic Bound States

[Function] double gsl_sf_hydrogenicR_1 (double Z, double r)

[Function] int gsl_sf_hydrogenicR_1_e (double Z, double r, gsl sf result *

   result)

These routines compute the lowest-order normalized hydrogenic bound state radial wavefunction R_{1} := 2Z\sqrt{Z} exp(−Zr).

[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} := \frac{2Z^{3/2}}{n^{2}} (\frac{2Zr}{n} )^{l} \sqrt{\frac{(n − l − 1)!}{(n + l)!}} exp(−Zr/n)L_{n−l−1}^{2l+1}(2Zr/n).

where L_{b}^{a}(x) is the generalized Laguerre polynomial (see Section 7.22 [Laguerre Functions], page 59). The normalization is chosen such that the wavefunction ψ is given by ψ(n, l, r) = R_{n}Y_{lm}.

7.7.2 Coulomb Wave Functions

The Coulomb wave functions F_{L}(η, x), G_{L}(η, 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}(η, x) = fc[k_{L}] ∗ exp(exp_{F})
G_{L}(η, x) = gc[k_{L}] ∗ exp(exp_{G})
F′_{L}(η, x) = fcp[k_{L}] ∗ exp(exp_{F})
G′_{L}(η, 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}(η, x), G_{L−k}(η, x) and their derivatives F′_{L}(η, x), G′_{L−k}(η, 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}(η, x) for L = Lmin … 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}(η, x), G_{L}(η, x) for L = Lmin … 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}(η, x), G_{L}(η, x) and their derivatives F′_{L}(η, x), G′_{L}(η, x) for L = Lmin … 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}(η, x)/x for L = Lmin … 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 η → 0.

7.7.3 Coulomb Wave Function Normalization Constant

The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7.

[Function] int gsl_sf_coulomb_CL_e (double L, double eta, gsl sf result *

   result)

This function computes the Coulomb wave function normalization constant C_{L}(η) 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}(η) for L = Lmin … Lmin + kmax, Lmin > −1.

7.8 Coupling Coefficients

The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for combined angular momentum vectors. Since the arguments of the standard coupling coefficient functions are integer or half-integer, the arguments of the following functions are, by convention, integers equal to twice the actual spin value. For information on the 3-j coefficients see Abramowitz & Stegun, Section 27.9. The functions described in this section are declared in the header

file ‘gsl_sf_coupling.h’.

7.8.1 3-j Symbols

[Function] double gsl_sf_coupling_3j (int two_ja, int two_jb, int two_jc, int
   two_ma, int two_mb, int two_mc)
[Function] int gsl_sf_coupling_3j_e (int two_ja, int two_jb, int two_jc, int
   two_ma, int two_mb, int two_mc, gsl sf result * result)

These routines compute the Wigner 3-j coefficient,

( \begin{matrix}ja & jb & jc \\ ma & mb & mc \end{matrix} )

where the arguments are given in half-integer units, ja = two ja/2, ma = two ma/2, etc.

7.8.2 6-j Symbols

[Function] double gsl_sf_coupling_6j (int two_ja, int two_jb, int two_jc, int
   two_jd, int two_je, int two_jf)
[Function] int gsl_sf_coupling_6j_e (int two_ja, int two_jb, int two_jc, int
   two_jd, int two_je, int two_jf, gsl sf result * result)

These routines compute the Wigner 6-j coefficient,

\{ \begin{matrix}ja & jb & jc \\ jd & je & jf \end{matrix} \}

where the arguments are given in half-integer units, ja = two ja/2, ma = two ma/2, etc.

7.8.3 9-j Symbols

[Function] double gsl_sf_coupling_9j (int two_ja, int two_jb, int two_jc, int
   two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji)
[Function] int gsl_sf_coupling_9j_e (int two_ja, int two_jb, int two_jc, int
   two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji,

gsl sf result * result) These routines compute the Wigner 9-j coefficient,

\{ \begin{matrix}ja & jb & jc \\ jd & je & jf \\ jg & jh & ji \end{matrix} \}

where the arguments are given in half-integer units, ja = two ja/2, ma = two ma/2, etc.

7.9 Dawson Function

The Dawson integral is defined by exp(−x^{2}) ∫_{0}^{x} dt exp(t^{2}). A table of Dawson’s integral can be found in Abramowitz & Stegun, Table 7.5. The Dawson functions are declared in the

header file ‘gsl_sf_dawson.h’.
[Function] double gsl_sf_dawson (double x)

[Function] int gsl_sf_dawson_e (double x, gsl sf result * result)

These routines compute the value of Dawson’s integral for x.

7.10 Debye Functions

The Debye functions D_{n}(x) are defined by the following integral,

D_{n}(x) = \frac{n}{x^{n}} ∫_{0}^{x} dt\frac{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) ∫_{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}) ∫_{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}) ∫_{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}) ∫_{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}) ∫_{0}^{x} dt(t^{5}/(e^{t} − 1)).

[Function] double gsl_sf_debye_6 (double x)

[Function] int gsl_sf_debye_6_e (double x, gsl sf result * result)

These routines compute the fourth-order Debye function D_{6}(x) = (6/x^{6}) ∫_{0}^{x} dt(t^{6}/(e^{t}− 1)).

7.11 Dilogarithm

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

7.11.1 Real Argument

[Function] double gsl_sf_dilog (double x)

[Function] int gsl_sf_dilog_e (double x, gsl sf result * result)

These routines compute the dilogarithm for a real argument. In Lewin’s notation this is Li_{2}(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_{2}(x) = −Re ∫_{0}^{x} ds log(1 − s)/s. Note that Im(Li_{2}(x)) = 0 for x ≤ 1, and −π log(x) for x > 1.

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(iθ). The real and imaginary parts of the result are returned in result re, result im.

7.12 Elementary Operations

The following functions allow for the propagation of errors when combining quantities by multiplication. The functions are declared in the header file ‘gsl_sf_elementary.h’.

[Function] 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.

7.13 Elliptic Integrals

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

7.13.1 Definition of Legendre Forms

The Legendre forms of elliptic integrals F (ϕ, k), E(ϕ, k) and P (ϕ, k, n) are defined by,

F (ϕ, k) = ∫_{0}^{ϕ} dt\frac{1}{\sqrt{(1 − k^{2} sin^{2}(t))}}

E(ϕ, k) = ∫_{0}^{ϕ} dt\sqrt{(1 − k^{2} sin^{2}(t))}
P (ϕ, k, n) = ∫_{0}^{ϕ} dt\frac{1}{(1 + n sin^{2}(t))\sqrt{1 − k^{2} sin^{2}(t)}}

The complete Legendre forms are denoted by K(k) = F (π/2, k) and E(k) = E(π/2, k). Further information on the Legendre forms of elliptic integrals can be found in Abramowitz & Stegun, Chapter 17. The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun.

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,

\begin{aligned}RC(x, y) & = 1/2 ∫_{0}^{∞} dt(t + x)^{−1/2}(t + y)^{−1} \\ RD(x, y, z) & = 3/2 ∫_{0}^{∞} dt(t + x)^{−1/2}(t + y)^{−1/2}(t + z)^{−3/2} \\ RF (x, y, z) & = 1/2 ∫_{0}^{∞} dt(t + x)^{−1/2}(t + y)^{−1/2}(t + z)^{−1/2} \\ RJ(x, y, z, p) & = 3/2 ∫_{0}^{∞} dt(t + x)^{−1/2}(t + y)^{−1/2}(t + z)^{−1/2}(t + p)^{−1} \end{aligned}

7.13.3 Legendre Form of Complete Elliptic Integrals

[Function] double gsl_sf_ellint_Kcomp (double k, gsl mode t mode)

[Function] int gsl_sf_ellint_Kcomp_e (double k, gsl mode t mode, gsl sf result

        * result)

These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode.

[Function] double gsl_sf_ellint_Ecomp (double k, gsl mode t mode)

[Function] int gsl_sf_ellint_Ecomp_e (double k, gsl mode t mode, gsl sf result

   * result)

These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode.

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 (ϕ, k) to the accuracy specified by the mode variable mode.

[Function] double gsl_sf_ellint_E (double phi, double k, gsl mode t mode)

[Function] int gsl_sf_ellint_E_e (double phi, double k, gsl mode t mode,

gsl sf result * result) These routines compute the incomplete elliptic integral E(ϕ, k) to the accuracy specified by the mode variable mode.

[Function] double gsl_sf_ellint_P (double phi, double k, double n, gsl mode t

   mode)

[Function] int gsl_sf_ellint_P_e (double phi, double k, double n, gsl mode t

mode, gsl sf result * result) These routines compute the incomplete elliptic integral P (ϕ, k, n) to the accuracy specified by the mode variable mode.

[Function] double gsl_sf_ellint_D (double phi, double k, double n, gsl mode t

   mode)

[Function] int gsl_sf_ellint_D_e (double phi, double k, double n, gsl mode t

mode, gsl sf result * result) These functions compute the incomplete elliptic integral D(ϕ, k, n) which is defined through the Carlson form RD(x, y, z) by the following relation,

D(ϕ, k, n) = RD(1 − sin^{2}(ϕ), 1 − k^{2} sin^{2}(ϕ), 1).

7.13.5 Carlson Forms

[Function] double gsl_sf_ellint_RC (double x, double y, gsl mode t mode)

[Function] int gsl_sf_ellint_RC_e (double x, double y, gsl mode t mode,

gsl sf result * result) These routines compute the incomplete elliptic integral RC(x, y) to the accuracy specified by the mode variable mode.

[Function] double gsl_sf_ellint_RD (double x, double y, double z, gsl mode t

   mode)

[Function] int gsl_sf_ellint_RD_e (double x, double y, double z, gsl mode t

mode, gsl sf result * result) These routines compute the incomplete elliptic integral RD(x, y, z) to the accuracy specified by the mode variable mode.

[Function] double gsl_sf_ellint_RF (double x, double y, double z, gsl mode t

   mode)

[Function] int gsl_sf_ellint_RF_e (double x, double y, double z, gsl mode t

mode, gsl sf result * result) These routines compute the incomplete elliptic integral RF (x, y, z) to the accuracy specified by the mode variable mode.

[Function] double gsl_sf_ellint_RJ (double x, double y, double z, double p, gsl mode t mode)

[Function] int gsl_sf_ellint_RJ_e (double x, double y, double z, double p,

gsl mode t mode, gsl sf result * result) These routines compute the incomplete elliptic integral RJ(x, y, z, p) to the accuracy specified by the mode variable mode.

7.14 Elliptic Functions (Jacobi)

The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16. The functions are declared in the header file ‘gsl_sf_elljac.h’.

[Function] int gsl_sf_elljac_e (double u, double m, double * sn, double * cn,

double * dn) This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations.

7.15 Error Functions

The error function is described in Abramowitz & Stegun, Chapter 7. The functions in this section are declared in the header file ‘gsl_sf_erf.h’.

7.15.1 Error Function

[Function] double gsl_sf_erf (double x)

[Function] int gsl_sf_erf_e (double x, gsl sf result * result)

These routines compute the error function erf(x), where erf(x) = (2/\sqrt{π}) ∫_{0}^{x} dt exp(−t^{2}).

7.15.2 Complementary Error Function

[Function] double gsl_sf_erfc (double x)

[Function] int gsl_sf_erfc_e (double x, gsl sf result * result)

These routines compute the complementary error function erfc(x) = 1 − erf(x) = (2/\sqrt{π}) ∫_{x}^{∞} exp(−t^{2}).

7.15.3 Log Complementary Error Function

[Function] double gsl_sf_log_erfc (double x)

[Function] int gsl_sf_log_erfc_e (double x, gsl sf result * result)

These routines compute the logarithm of the complementary error function log(erfc(x)).

7.15.4 Probability functions

The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.

[Function] double gsl_sf_erf_Z (double x)

[Function] int gsl_sf_erf_Z_e (double x, gsl sf result * result)

These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2π}) 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{2π}) ∫_{x}^{∞} 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) = \frac{Z(x)}{Q(x)} = \sqrt{\frac{2}{π}} \frac{exp(−x^{2}/2)}{erfc(x/\sqrt{2})}

It decreases rapidly as x approaches −∞ and asymptotes to h(x) ∼ x as x approaches +∞.

[Function] double gsl_sf_hazard (double x)

[Function] int gsl_sf_hazard_e (double x, gsl sf result * result)

These routines compute the hazard function for the normal distribution.

7.16 Exponential Functions

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

7.16.1 Exponential Function

[Function] double gsl_sf_exp (double x)

[Function] int gsl_sf_exp_e (double x, gsl sf result * result)

These routines provide an exponential function exp(x) using GSL semantics and error checking.

[Function] int gsl_sf_exp_e10_e (double x, gsl sf result e10 * result)

This function computes the exponential exp(x) using the gsl_sf_result_e10 type to return a result with extended range. This function may be useful if the value of exp(x) would overflow the numeric range of double.

[Function] double gsl_sf_exp_mult (double x, double y)

[Function] int gsl_sf_exp_mult_e (double x, double y, gsl sf result * result)

These routines exponentiate x and multiply by the factor y to return the product y exp(x).

[Function] int gsl_sf_exp_mult_e10_e (const double x, const double y,

gsl sf result e10 * result) This function computes the product y exp(x) using the gsl_sf_result_e10 type to return a result with extended numeric range.

7.16.2 Relative Exponential Functions

[Function] double gsl_sf_expm1 (double x)

[Function] int gsl_sf_expm1_e (double x, gsl sf result * result)

These routines compute the quantity exp(x) − 1 using an algorithm that is accurate for small x.

[Function] double gsl_sf_exprel (double x)

[Function] int gsl_sf_exprel_e (double x, gsl sf result * result)

These routines compute the quantity (exp(x) − 1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (exp(x)−1)/x = 1 + x/2 + x^{2}/(2 ∗ 3) + x^{3}/(2 ∗ 3 ∗ 4) + . . ..

[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) + . . ..

[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) − ∑_{k=0}^{N−1} x^{k}/k!)
= 1 + x/(N + 1) + x^{2}/((N + 1)(N + 2)) + …
= _{1}F_{1}(1, 1 + N, x)

7.16.3 Exponentiation With Error Estimate

[Function] int gsl_sf_exp_err_e (double x, double dx, gsl sf result * result)

This function exponentiates x with an associated absolute error dx.

[Function] int gsl_sf_exp_err_e10_e (double x, double dx, gsl sf result e10 *

   result)

This function exponentiates a quantity x with an associated absolute error dx using the gsl_sf_result_e10 type to return a result with extended range.

[Function] int gsl_sf_exp_mult_err_e (double x, double dx, double y, double

dy, gsl sf result * result) This routine computes the product y exp(x) for the quantities x, y with associated absolute errors dx, dy.

[Function] int gsl_sf_exp_mult_err_e10_e (double x, double dx, double y,

double dy, gsl sf result e10 * result) This routine computes the product y exp(x) for the quantities x, y with associated absolute errors dx, dy using the gsl_sf_result_e10 type to return a result with extended range.

7.17 Exponential Integrals

Information on the exponential integrals can be found in Abramowitz & Stegun, Chapter 5. These functions are declared in the header file ‘gsl_sf_expint.h’.

7.17.1 Exponential Integral

[Function] double gsl_sf_expint_E1 (double x)

[Function] int gsl_sf_expint_E1_e (double x, gsl sf result * result)

These routines compute the exponential integral E_{1}(x),

E_{1}(x) := Re ∫_{1}^{∞} 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 ∫_{1}^{∞} dt exp(−xt)/t^{2}.

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) := −P V (∫_{−x}^{∞} dt exp(−t)/t)

where P V denotes the principal value of the integral.

7.17.3 Hyperbolic Integrals

[Function] double gsl_sf_Shi (double x)

[Function] int gsl_sf_Shi_e (double x, gsl sf result * result)

These routines compute the integral Shi(x) = ∫_{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[γ_{E} + log(x) + ∫_{0}^{x} dt(cosh[t] − 1)/t], where γ_{E} is the Euler constant (available as the macro M_EULER).

7.17.4 Ei 3(x)

[Function] double gsl_sf_expint_3 (double x)

[Function] int gsl_sf_expint_3_e (double x, gsl sf result * result)

These routines compute the third-order exponential integral Ei_{3}(x) = ∫_{0}^{x} dt exp(−t^{3}) for x ≥ 0.

7.17.5 Trigonometric Integrals

[Function] double gsl_sf_Si (const double x)

[Function] int gsl_sf_Si_e (double x, gsl sf result * result)

These routines compute the Sine integral Si(x) = ∫_{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) = − ∫_{x}^{∞} dt cos(t)/t for x > 0.

7.17.6 Arctangent Integral

[Function] double gsl_sf_atanint (double x)

[Function] int gsl_sf_atanint_e (double x, gsl sf result * result)

These routines compute the Arctangent integral, which is defined as AtanInt(x) = ∫_{0}^{x} dt arctan(t)/t.

7.18 Fermi-Dirac Function

The functions described in this section are declared in the header file

‘gsl_sf_fermi_dirac.h’.

7.18.1 Complete Fermi-Dirac Integrals

The complete Fermi-Dirac integral F_{j}(x) is given by,

F_{j}(x) := \frac{1}{Γ(j + 1)} ∫_{0}^{∞} dt\frac{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) = ∫_{0}^{∞} 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) ∫_{0}^{∞} 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/Γ(j + 1)) ∫_{0}^{∞} dt(t^{j}/(exp(t − x) + 1)).

[Function] double gsl_sf_fermi_dirac_mhalf (double x)
[Function] int gsl_sf_fermi_dirac_mhalf_e (double x, gsl sf result * result)

These routines compute the complete Fermi-Dirac integral F_{−1/2}(x).

[Function] double gsl_sf_fermi_dirac_half (double x)
[Function] int gsl_sf_fermi_dirac_half_e (double x, gsl sf result * result)

These routines compute the complete Fermi-Dirac integral F_{1/2}(x).

[Function] double gsl_sf_fermi_dirac_3half (double x)
[Function] int gsl_sf_fermi_dirac_3half_e (double x, gsl sf result * result)

These routines compute the complete Fermi-Dirac integral F_{3/2}(x).

7.18.2 Incomplete Fermi-Dirac Integrals

The incomplete Fermi-Dirac integral F_{j}(x, b) is given by,

F_{j}(x, b) := \frac{1}{Γ(j + 1)} ∫_{b}^{∞} dt\frac{t^{j}}{(exp(t − x) + 1)}

[Function] double gsl_sf_fermi_dirac_inc_0 (double x, double b)

[Function] int gsl_sf_fermi_dirac_inc_0_e (double x, double b, gsl sf result *

   result)

These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_{0}(x, b) = ln(1 + e^{b−x}) − (b − x).

7.19 Gamma and Beta Functions

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

7.19.1 Gamma Functions

The Gamma function is defined by the following integral,

Γ(x) = ∫_{0}^{∞} dt t^{x−1} exp(−t)

It is related to the factorial function by Γ(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 Γ(x), subject to x not being a negative integer. The function is computed using the real Lanczos method. The maximum value of x such that Γ(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(Γ(x)), subject to x not a being negative integer. For x < 0 the real part of log(Γ(x)) is returned, which is equivalent to log(|Γ(x)|). The function is computed using the real Lanczos method.

[Function] int gsl_sf_lngamma_sgn_e (double x, gsl sf result * result_lg,

double * sgn) This routine computes the sign of the gamma function and the logarithm of its magnitude, subject to x not being a negative integer. The function is computed using the real Lanczos method. The value of the gamma function can be reconstructed using the relation Γ(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 Γ^{∗}(x) for x > 0. The regulated gamma function is given by,

Γ^{∗}(x) = Γ(x)/(\sqrt{2π}x^{(x−1/2)} exp(−x))

= (1 + \frac{1}{12x} + ...) for x → ∞

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/Γ(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(Γ(z)) for complex z = z_{r} +iz_{i} and z not a negative integer, using the complex Lanczos method. The returned parameters are lnr = log |Γ(z)| and arg = arg(Γ(z)) in (−π, π]. Note that the phase part (arg) is not well-determined when |z| is very large, due to inevitable roundoff in restricting to (−π, π]. This will result in a GSL_ELOSS error when it occurs. The absolute value part (lnr), however, never suffers from loss of precision.

7.19.2 Factorials

Although factorials can be computed from the Gamma function, using the relation n! = Γ(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! = Γ(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) . . .. 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(Γ(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.

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

[Function] double gsl_sf_lnpoch (double a, double x)

[Function] int gsl_sf_lnpoch_e (double a, double x, gsl sf result * result)

These routines compute the logarithm of the Pochhammer symbol, log((a)_{x}) = log(Γ(a + x)/Γ(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} = Γ(a + x)/Γ(a), subject to a, a + x not being negative integers.

[Function] double gsl_sf_pochrel (double a, double x)

[Function] int gsl_sf_pochrel_e (double a, double x, gsl sf result * result)

These routines compute the relative Pochhammer symbol ((a)_{x} − 1)/x where (a)_{x} = Γ(a + x)/Γ(a).

7.19.4 Incomplete Gamma Functions

[Function] double gsl_sf_gamma_inc (double a, double x)

[Function] int gsl_sf_gamma_inc_e (double a, double x, gsl sf result * result)

These functions compute the unnormalized incomplete Gamma Function Γ(a, x) = ∫_{x}^{∞} 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/Γ(a) ∫_{x}^{∞} 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/Γ(a) ∫_{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).

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) = Γ(a)Γ(b)/Γ(a + b) for a > 0, b > 0.

[Function] double gsl_sf_lnbeta (double a, double b)

[Function] int gsl_sf_lnbeta_e (double a, double b, gsl sf result * result)

These routines compute the logarithm of the Beta Function, log(B(a, b)) for a > 0, b > 0.

7.19.6 Incomplete Beta Function

[Function] double gsl_sf_beta_inc (double a, double b, double x)

[Function] int gsl_sf_beta_inc_e (double a, double b, double x, gsl sf result *

   result)

These routines compute the normalized incomplete Beta function B_{x}(a, b)/B(a, b) where B_{x}(a, b) = ∫_{0}^{x} t^{a−1}(1 − t)^{b−1}dt for a > 0, b > 0, and 0 ≤ x ≤ 1.

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_{n}^{(λ)}(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_{n}^{(λ)}(x) for a specific value of n, lambda, x subject to λ > −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_{n}^{(λ)}(x) for n = 0, 1, 2, …, nmax, subject to λ > −1/2, nmax ≥ 0.

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 _{0}F_{1}(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 _{1}F_{1}(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 _{1}F_{1}(a, b, x) = M(a, b, x) for general parameters a, b.

[Function] double gsl_sf_hyperg_U_int (int m, int n, double x)

[Function] int gsl_sf_hyperg_U_int_e (int m, int n, double x, gsl sf result *

   result)

These routines compute the confluent hypergeometric function U(m, n, x) for integer parameters m, n.

[Function] int gsl_sf_hyperg_U_int_e10_e (int m, int n, double x,

gsl sf result e10 * result) This routine computes the confluent hypergeometric function U(m, n, x) for integer parameters m, n using the gsl_sf_result_e10 type to return a result with extended range.

[Function] double gsl_sf_hyperg_U (double a, double b, double x)

[Function] int gsl_sf_hyperg_U_e (double a, double b, double x)

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

[Function] int gsl_sf_hyperg_U_e10_e (double a, double b, double x,

gsl sf result e10 * result) This routine computes the confluent hypergeometric function U(a, b, x) using the gsl_sf_result_e10 type to return a result with extended range.

[Function] double gsl_sf_hyperg_2F1 (double a, double b, double c, double x)

[Function] int gsl_sf_hyperg_2F1_e (double a, double b, double c, double x,

gsl sf result * result) These routines compute the Gauss hypergeometric function _{2}F_{1}(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_{2}F_{1}(a_{R}+ia_{I}, aR−iaI, 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 2F_{1}(a, b, c, x)/Γ(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 _{2}F_{1}(a_{R} + ia_{I}, a_{R} − ia_{I}, c, x)/Γ(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 _{2}F_{0}(a, b, x). The series representation is a divergent hypergeometric series. However, for x < 0 we have _{2}F_{0}(a, b, x) = (−1/x)^{a}U(a, 1 + a − b, −1/x)

7.22 Laguerre Functions

The generalized Laguerre polynomials are defined in terms of confluent hypergeometric functions as L_{n}^{a}(x) = ((a + 1)_{n}/n!)_{1}F_{1}(−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_{n}^{0}(x) = L_{n}(x) and L_{n}^{k}(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_{1}^{a}(x), L_{2}^{a}(x), L_{3}^{a}(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_{n}^{a}(x) for a > −1, n ≥ 0.

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).

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’.

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, …, 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.

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)/(4π)}\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)/(4π)}\sqrt{(l − m)!/(l + m)!}P_{l}^{m}(x), and optionally their derivatives, for m ≥ 0, l = |m|, …, lmax, |x| ≤ 1

[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.

7.24.3 Conical Functions

The Conical Functions P_{−(1/2)+iλ}^{μ}(x) and Q_{−(1/2)+iλ}^{μ} 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+iλ}^{1/2}(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+iλ}^{−1/2}(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_{−1/2+iλ}^{0}(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_{−1/2+iλ}^{1}(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+iλ}^{−1/2−l}(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_{−1/2+iλ}^{−m}(x) for x > −1, m ≥ −1.

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, λ → ∞, η → 0, λη 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 3dimensional hyperbolic space,

L_{0}^{H3d}(λ, η) := \frac{sin(λη)}{λ sinh(η)}

for η ≥ 0. In the flat limit this takes the form L_{0}^{H3d}(λ, η) = j_{0}(λη).

[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 3dimensional hyperbolic space,

L_{1}^{H3d}(λ, η) := \frac{1}{\sqrt{λ^{2} + 1}}(\frac{sin(λη)}{λ sinh(η)}) (coth(η) − λ cot(λη))

for η ≥ 0. In the flat limit this takes the form L_{1}^{H3d}(λ, η) = j_{1}(λη).

[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 3dimensional hyperbolic space η ≥ 0, l ≥ 0. In the flat limit this takes the form L_{l}^{H3d}(λ, η) = j_{l}(λη).

[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_{l}^{H3d}(λ, η) for 0 ≤ l ≤ lmax.

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 ≠ 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} +iz_{i}. The results are returned as lnr, theta such that exp(lnr + iθ) = z_{r} + iz_{i}, where θ lies in the range [−π, π].

[Function] double gsl_sf_log_1plusx (double x)

[Function] int gsl_sf_log_1plusx_e (double x, gsl sf result * result)

These routines compute log(1 + x) for x > −1 using an algorithm that is accurate for small x.

[Function] double gsl_sf_log_1plusx_mx (double x)
[Function] int gsl_sf_log_1plusx_mx_e (double x, gsl sf result * result)

These routines compute log(1 + x) − x for x > −1 using an algorithm that is accurate for small x.

7.26 Power Function

The following functions are equivalent to the function gsl_pow_int (see Section 4.4 [Small integer powers], page 18) with an error estimate. These functions are declared in the header

file ‘gsl_sf_pow_int.h’.
[Function] double gsl_sf_pow_int (double x, int n)

[Function] int gsl_sf_pow_int_e (double x, int n, gsl sf result * result)

These routines compute the power x^{n} for integer n. The power is computed using the minimum number of multiplications. For example, x^{8} is computed as ((x^{2})^{2})^{2}, requiring only 3 multiplications. For reasons of efficiency, these functions do not check for overflow or underflow conditions.

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

7.27 Psi (Digamma) Function

The polygamma functions of order m are defined by

ψ^{(m)}(x) = (\frac{d}{dx})^{m} ψ(x) = (\frac{d}{dx})^{m+1} log(Γ(x))

where ψ(x) = Γ′(x)/Γ(x) is known as the digamma function. These functions are declared in the header file ‘gsl_sf_psi.h’.

7.27.1 Digamma Function

[Function] double gsl_sf_psi_int (int n)

[Function] int gsl_sf_psi_int_e (int n, gsl sf result * result)

These routines compute the digamma function ψ(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 ψ(x) for general x, x ≠ 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 + iy, Re[ψ(1 + iy)].

7.27.2 Trigamma Function

[Function] double gsl_sf_psi_1_int (int n)
[Function] int gsl_sf_psi_1_int_e (int n, gsl sf result * result)

These routines compute the Trigamma function ψ′(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 ψ′(x) for general x.

7.27.3 Polygamma Function

[Function] double gsl_sf_psi_n (int m, double x)

[Function] int gsl_sf_psi_n_e (int m, double x, gsl sf result * result)

These routines compute the polygamma function ψ^{(m)}(x) for m ≥ 0, x > 0.

7.28 Synchrotron Functions

The functions described in this section are declared in the header file

‘gsl_sf_synchrotron.h’.

[Function] double gsl_sf_synchrotron_1 (double x)
[Function] int gsl_sf_synchrotron_1_e (double x, gsl sf result * result)

These routines compute the first synchrotron function x ∫_{x}^{∞} dtK_{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 xK_{2/3}(x) for x ≥ 0.

7.29 Transport Functions

The transport functions J(n, x) are defined by the integral representations J(n, x) := ∫_{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).

7.30 Trigonometric Functions

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

‘gsl_sf_trig.h’.

7.30.1 Circular Trigonometric Functions

[Function] double gsl_sf_sin (double x)

[Function] int gsl_sf_sin_e (double x, gsl sf result * result)

These routines compute the sine function sin(x).

[Function] double gsl_sf_cos (double x)

[Function] int gsl_sf_cos_e (double x, gsl sf result * result)

These routines compute the cosine function cos(x).

[Function] double gsl_sf_hypot (double x, double y)

[Function] int gsl_sf_hypot_e (double x, double y, gsl sf result * result)

These routines compute the hypotenuse function \sqrt{x^{2} + y^{2}} avoiding overflow and underflow.

[Function] double gsl_sf_sinc (double x)

[Function] int gsl_sf_sinc_e (double x, gsl sf result * result)

These routines compute sinc(x) = sin(πx)/(πx) for any value of x.

7.30.2 Trigonometric Functions for Complex Arguments

[Function] int gsl_sf_complex_sin_e (double zr, double zi, gsl sf result * szr,

gsl sf result * szi) This function computes the complex sine, sin(z_{r} + iz_{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}+iz_{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} + iz_{i})) storing the real and imaginary parts in szr, szi.

7.30.3 Hyperbolic Trigonometric Functions

[Function] double gsl_sf_lnsinh (double x)

[Function] int gsl_sf_lnsinh_e (double x, gsl sf result * result)

These routines compute log(sinh(x)) for x > 0.

[Function] double gsl_sf_lncosh (double x)

[Function] int gsl_sf_lncosh_e (double x, gsl sf result * result)

These routines compute log(cosh(x)) for any x.

7.30.4 Conversion Functions

[Function] int gsl_sf_polar_to_rect (double r, double theta, gsl sf result * x,

gsl sf result * y); This function converts the polar coordinates (r,theta) to rectilinear coordinates (x,y), x = r cos(θ), y = r sin(θ).

[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(θ), y = r sin(θ). The argument theta lies in the range [−π, π].

7.30.5 Restriction Functions

[Function] double gsl_sf_angle_restrict_symm (double theta)
[Function] int gsl_sf_angle_restrict_symm_e (double * theta)

These routines force the angle theta to lie in the range (−π, π].

[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, 2π).

7.30.6 Trigonometric Functions With Error Estimates

[Function] int gsl_sf_sin_err_e (double x, double dx, gsl sf result * result)

This routine computes the sine of an angle x with an associated absolute error dx, sin(x ± 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 ± dx). Note that this function is provided in the error-handling form only since its purpose is to compute the propagated error.

7.31 Zeta Functions

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

7.31.1 Riemann Zeta Function

The Riemann zeta function is defined by the infinite sum ζ(s) = ∑_{k=1}^{∞} 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 ζ(n) for integer n, n ≠ 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 ζ(s) for arbitrary s, s ≠ 1.

7.31.2 Riemann Zeta Function Minus One

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

[Function] double gsl_sf_zetam1_int (int n)
[Function] int gsl_sf_zetam1_int_e (int n, gsl sf result * result)

These routines compute ζ(n) − 1 for integer n, n ≠ 1.

[Function] double gsl_sf_zetam1 (double s)

[Function] int gsl_sf_zetam1_e (double s, gsl sf result * result)

These routines compute ζ(s) − 1 for arbitrary s, s ≠ 1.

7.31.3 Hurwitz Zeta Function

The Hurwitz zeta function is defined by ζ(s, q) = ∑_{0}^{∞}(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 ζ(s, q) for s > 1, q > 0.

7.31.4 Eta Function

The eta function is defined by η(s) = (1 − 2^{1−s})ζ(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 η(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 η(s) for arbitrary s.

7.32 Examples

The following example demonstrates the use of the error handling form of the special functions, in this case to compute the Bessel function J_{0}(5.0),

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_bessel.h>
int
main (void)
{
  double x = 5.0;
  gsl_sf_result result;
  double expected = -0.17759677131433830434739701;
  int status = gsl_sf_bessel_J0_e (x, &result);
  printf ("status  = %s\n", gsl_strerror(status));
  printf ("J0(5.0) = %.18f\n"
          "      +/- % .18f\n",
          result.val, result.err);
  printf ("exact   = %.18f\n", expected);
  return status;
}

Here are the results of running the program,

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

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

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

 int
 main (void)
 {
   double x = 5.0;
   double expected = -0.17759677131433830434739701;
   double y = gsl_sf_bessel_J0 (x);
   printf ("J0(5.0) = %.18f\n", y);
   printf ("exact   = %.18f\n", expected);
   return 0;
 }

The results of the function are the same,

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

7.33 References and Further Reading

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

The following papers contain information on the algorithms used to compute the special functions,

MISCFUN: A software package to compute uncommon special functions. ACM Trans. Math. Soft., vol. 22, 1996, 288–301 G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edition (Cambridge University Press, 1944). G. Nemeth, Mathematical Approximations of Special Functions, Nova Science Publishers, ISBN 1-56072-052-2 B.C. Carlson, Special Functions of Applied Mathematics (1977) W.J. Thompson, Atlas for Computing Mathematical Functions, John Wiley & Sons, New York (1997). Y.Y. Luke, Algorithms for the Computation of Mathematical Functions, Academic Press, New York (1977).

8 Vectors and Matrices

The functions described in this chapter provide a simple vector and matrix interface to ordinary C arrays. The memory management of these arrays is implemented using a single underlying type, known as a block. By writing your functions in terms of vectors and matrices you can pass a single structure containing both data and dimensions as an argument without needing additional function parameters. The structures are compatible with the vector and matrix formats used by blas routines.

8.1 Data types

All the functions are available for each of the standard data-types. The versions for double have the prefix gsl_block, gsl_vector and gsl_matrix. Similarly the versions for single-

precision float arrays have the prefix gsl_block_float, gsl_vector_float and gsl_

matrix_float. The full list of available types is given below,

     gsl_block                       double
     gsl_block_float                 float
     gsl_block_long_double           long double
     gsl_block_int                   int
     gsl_block_uint                  unsigned int
     gsl_block_long                  long
     gsl_block_ulong                 unsigned long
     gsl_block_short                 short
     gsl_block_ushort                unsigned short
     gsl_block_char                  char
     gsl_block_uchar                 unsigned char
     gsl_block_complex               complex double
     gsl_block_complex_float         complex float
     gsl_block_complex_long_double   complex long double

Corresponding types exist for the gsl_vector and gsl_matrix functions.

8.2 Blocks

For consistency all memory is allocated through a gsl_block structure. The structure contains two components, the size of an area of memory and a pointer to the memory. The gsl_block structure looks like this,

     typedef struct
     {
       size_t size;
       double * data;
     } gsl_block;

Vectors and matrices are made by slicing an underlying block. A slice is a set of elements formed from an initial offset and a combination of indices and step-sizes. In the case of a matrix the step-size for the column index represents the row-length. The step-size for a vector is known as the stride.

The functions for allocating and deallocating blocks are defined in ‘gsl_block.h’

8.2.1 Block allocation

The functions for allocating memory to a block follow the style of malloc and free. In addition they also perform their own error checking. If there is insufficient memory available to allocate a block then the functions call the GSL error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn’t necessary to check every alloc.

[Function] gsl_block * gsl_block_alloc (size t n)

This function allocates memory for a block of n double-precision elements, returning a pointer to the block struct. The block is not initialized and so the values of its elements are undefined. Use the function gsl_block_calloc if you want to ensure that all the elements are initialized to zero. A null pointer is returned if insufficient memory is available to create the block.

[Function] gsl_block * gsl_block_calloc (size t n)

This function allocates memory for a block and initializes all the elements of the block to zero.

[Function] void gsl_block_free (gsl block * b)

This function frees the memory used by a block b previously allocated with gsl_

block_alloc or gsl_block_calloc.

8.2.2 Reading and writing blocks

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

[Function] int gsl_block_fwrite (FILE * stream, const gsl block * b)

This function writes the elements of the block b to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

[Function] int gsl_block_fread (FILE * stream, gsl block * b)

This function reads into the block b from the open stream stream in binary format. The block b must be preallocated with the correct length since the function uses the size of b to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

[Function] int gsl_block_fprintf (FILE * stream, const gsl block * b, const

   char * format)

This function writes the elements of the block b line-by-line to the stream stream using the format specifier format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.

[Function] int gsl_block_fscanf (FILE * stream, gsl block * b)

This function reads formatted data from the stream stream into the block b. The block b must be preallocated with the correct length since the function uses the size of b to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

8.2.3 Example programs for blocks

The following program shows how to allocate a block,

     #include <stdio.h>
     #include <gsl/gsl_block.h>
     int
     main (void)
     {
       gsl_block * b = gsl_block_alloc (100);
       printf ("length of block = %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

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’

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.

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.

8.3.3 Initializing vector elements

[Function] void gsl_vector_set_all (gsl vector * v, double x)

This function sets all the elements of the vector v to the value x.

[Function] void gsl_vector_set_zero (gsl vector * v)

This function sets all the elements of the vector v to zero.

[Function] int gsl_vector_set_basis (gsl vector * v, size t i)

This function makes a basis vector by setting all the elements of the vector v to zero except for the i-th element which is set to one.

8.3.4 Reading and writing vectors

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

[Function] int gsl_vector_fwrite (FILE * stream, const gsl vector * v)

This function writes the elements of the vector v to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

[Function] int gsl_vector_fread (FILE * stream, gsl vector * v)

This function reads into the vector v from the open stream stream in binary format. The vector v must be preallocated with the correct length since the function uses the size of v to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

[Function] int gsl_vector_fprintf (FILE * stream, const gsl vector * v, const

   char * format)

This function writes the elements of the vector v line-by-line to the stream stream using the format specifier format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.

[Function] int gsl_vector_fscanf (FILE * stream, gsl vector * v)

This function reads formatted data from the stream stream into the vector v. The vector v must be preallocated with the correct length since the function uses the size of v to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

8.3.5 Vector views

In addition to creating vectors from slices of blocks it is also possible to slice vectors and create vector views. For example, a subvector of another vector can be described with a view, or two views can be made which provide access to the even and odd elements of a vector.

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 nonconstant vectors, using separate types that preserve constness. A vector view has the type

gsl_vector_view and a constant vector view has the type gsl_vector_const_view. In

both cases the elements of the view can be accessed as a gsl_vector using the vector component of the view object. A pointer to a vector of type gsl_vector * or const gsl_ vector * can be obtained by taking the address of this component with the & operator.

When using this pointer it is important to ensure that the view itself remains in scope— the simplest way to do so is by always writing the pointer as &view.vector, and never storing this value in another variable.

[Function] gsl_vector_view gsl_vector_subvector (gsl vector * v, size t
   offset, size t n)
[Function] gsl_vector_const_view gsl_vector_const_subvector (const

gsl vector * v, size t offset, size t n) These functions return a vector view of a subvector of another vector v. The start of the new vector is offset by offset elements from the start of the original vector. The new vector has n elements. Mathematically, the i-th element of the new vector v’ is given by,

     v'(i) = v->data[(offset + i)*v->stride]

where the index i runs from 0 to n-1. The data pointer of the returned vector struct is set to null if the combined parameters (offset,n) overrun the end of the original vector. The new vector is only a view of the block underlying the original vector, v. The block containing the elements of v is not owned by the new vector. When the view goes out of scope the original vector v and its block will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.

The function gsl_vector_const_subvector is equivalent to gsl_vector_subvector

but can be used for vectors which are declared const.

[Function] gsl_vector_view gsl_vector_subvector_with_stride (gsl vector

* v, size t offset, size t stride, size t n)

[Function] gsl_vector_const_view
   gsl_vector_const_subvector_with_stride (const gsl vector * v, size t
   offset, size t stride, size t n)

These functions return a vector view of a subvector of another vector v with an additional stride argument. The subvector is formed in the same way as for gsl_ vector_subvector but the new vector has n elements with a step-size of stride from one element to the next in the original vector. Mathematically, the i-th element of the new vector v’ is given by,

     v'(i) = v->data[(offset + i*stride)*v->stride]

where the index i runs from 0 to n-1. Note that subvector views give direct access to the underlying elements of the original vector. For example, the following code will zero the even elements of the vector v of length n, while leaving the odd elements untouched,

     gsl_vector_view v_even
       = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
     gsl_vector_set_zero (&v_even.vector);

A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector would be, using &view.vector. For example, the following code computes the norm of the odd elements of v using the blas routine dnrm2,

     gsl_vector_view v_odd
       = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
     double r = gsl_blas_dnrm2 (&v_odd.vector);
The function gsl_vector_const_subvector_with_stride is equivalent to
gsl_vector_subvector_with_stride but can be used for vectors which are

declared const.

[Function] gsl_vector_view gsl_vector_complex_real (gsl vector complex *
   v)
[Function] gsl_vector_const_view gsl_vector_complex_const_real (const

gsl vector complex * v) These functions return a vector view of the real parts of the complex vector v.

The function gsl_vector_complex_const_real is equivalent to gsl_vector_

complex_real but can be used for vectors which are declared const.

[Function] gsl_vector_view gsl_vector_complex_imag (gsl vector complex *
   v)
[Function] gsl_vector_const_view gsl_vector_complex_const_imag (const

gsl vector complex * v) These functions return a vector view of the imaginary parts of the complex vector v.

The function gsl_vector_complex_const_imag is equivalent to gsl_vector_

complex_imag but can be used for vectors which are declared const.

[Function] gsl_vector_view gsl_vector_view_array (double * base, size t n)
[Function] gsl_vector_const_view gsl_vector_const_view_array (const

double * base, size t n) These functions return a vector view of an array. The start of the new vector is given by base and has n elements. Mathematically, the i-th element of the new vector v’ is given by,

     v'(i) = base[i]

where the index i runs from 0 to n-1. The array containing the elements of v is not owned by the new vector view. When the view goes out of scope the original array will continue to exist. The original memory can only be deallocated by freeing the original pointer base. Of course, the original array should not be deallocated while the view is still in use.

The function gsl_vector_const_view_array is equivalent to gsl_vector_view_

array but can be used for arrays which are declared const.

[Function] gsl_vector_view gsl_vector_view_array_with_stride (double *

base, size t stride, size t n)

[Function] gsl_vector_const_view
   gsl_vector_const_view_array_with_stride (const double * base,

size t stride, size t n) These functions return a vector view of an array base with an additional stride argument. The subvector is formed in the same way as for gsl_vector_view_array but the new vector has n elements with a step-size of stride from one element to the next in the original array. Mathematically, the i-th element of the new vector v’ is given by,

     v'(i) = base[i*stride]

where the index i runs from 0 to n-1. Note that the view gives direct access to the underlying elements of the original array. A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector would be, using &view.vector.

The function gsl_vector_const_view_array_with_stride is equivalent to gsl_

vector_view_array_with_stride but can be used for arrays which are declared

const.

8.3.6 Copying vectors

Common operations on vectors such as addition and multiplication are available in the blas part of the library (see Chapter 12 [BLAS Support], page 110). However, it is useful to have a small number of utility functions which do not require the full blas code. The following functions fall into this category.

[Function] int gsl_vector_memcpy (gsl vector * dest, const gsl vector * src)

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

[Function] int gsl_vector_swap (gsl vector * v, gsl vector * w)

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

8.3.7 Exchanging elements

The following 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.

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} = xa_{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.

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.

8.3.10 Vector properties

[Function] int gsl_vector_isnull (const gsl vector * v)

This function returns 1 if all the elements of the vector v are zero, and 0 otherwise.

8.3.11 Example programs for vectors

This program shows how to allocate, initialize and read from a vector using the functions

gsl_vector_alloc, gsl_vector_set and gsl_vector_get.
#include <stdio.h>
#include <gsl/gsl_vector.h>
int
main (void)
{
  int i;
  gsl_vector * v = gsl_vector_alloc (3);
  for (i = 0; i < 3; i++)
    {
      gsl_vector_set (v, i, 1.23 + i);
    }
  for (i = 0; i < 100; i++)
    {
      printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
    }
  return 0;
}

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

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

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

#include <stdio.h>
#include <gsl/gsl_vector.h>
int
main (void)
{
  int i;

gsl_vector * v = gsl_vector_alloc (100);
for (i = 0; i < 100; i++)
  {
    gsl_vector_set (v, i, 1.23 + i);
  }
{
   FILE * f = fopen ("test.dat", "w");
   gsl_vector_fprintf (f, v, "%.5g");
   fclose (f);
}
return 0;
}

After running this program the file ‘test.dat’ should contain the elements of v, written using the format specifier %.5g. The vector could then be read back in using the function

gsl_vector_fscanf (f, v) as follows:
#include <stdio.h>
#include <gsl/gsl_vector.h>
int
main (void)
{
int i;
gsl_vector * v = gsl_vector_alloc (10);
{
   FILE * f = fopen ("test.dat", "r");
   gsl_vector_fscanf (f, v);
   fclose (f);
}
for (i = 0; i < 10; i++)
  {
    printf ("%g\n", gsl_vector_get(v, i));
  }
return 0;
}

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’

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.

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.

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) = δ(i, j), i.e. a unit diagonal with all off-diagonal elements zero. This applies to both square and rectangular matrices.

8.4.4 Reading and writing matrices

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

[Function] int gsl_matrix_fwrite (FILE * stream, const gsl matrix * m)

This function writes the elements of the matrix m to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

[Function] int gsl_matrix_fread (FILE * stream, gsl matrix * m)

This function reads into the matrix m from the open stream stream in binary format. The matrix m must be preallocated with the correct dimensions since the function uses the size of m to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

[Function] int gsl_matrix_fprintf (FILE * stream, const gsl matrix * m, const

   char * format)

This function writes the elements of the matrix m line-by-line to the stream stream using the format specifier format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.

[Function] int gsl_matrix_fscanf (FILE * stream, gsl matrix * m)

This function reads formatted data from the stream stream into the matrix m. The matrix m must be preallocated with the correct dimensions since the function uses the size of m to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

8.4.5 Matrix views

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 nonconstant matrices using separate types that preserve constness. A matrix view has the type

gsl_matrix_view and a constant matrix view has the type gsl_matrix_const_view. In

both cases the elements of the view can by accessed using the matrix component of the view object. A pointer gsl_matrix * or const gsl_matrix * can be obtained by taking the address of the matrix component with the & operator. In addition to matrix views it is also possible to create vector views of a matrix, such as row or column views.

[Function] gsl_matrix_view gsl_matrix_submatrix (gsl matrix * m, size t k1,

size t k2, size t n1, size t n2)

[Function] gsl_matrix_const_view gsl_matrix_const_submatrix (const

gsl matrix * m, size t k1, size t k2, size t n1, size t n2) These functions return a matrix view of a submatrix of the matrix m. The upper-left element of the submatrix is the element (k1,k2) of the original matrix. The submatrix has n1 rows and n2 columns. The physical number of columns in memory given by tda is unchanged. Mathematically, the (i, j)-th element of the new matrix is given by,

     m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1. The data pointer of the returned matrix struct is set to null if the combined parameters (i,j,n1,n2,tda) overrun the ends of the original matrix. The new matrix view is only a view of the block underlying the existing matrix, m. The block containing the elements of m is not owned by the new matrix view. When the view goes out of scope the original matrix m and its block will continue to exist. The original memory can only be deallocated by freeing the original matrix. Of course, the original matrix should not be deallocated while the view is still in use.

The function gsl_matrix_const_submatrix is equivalent to gsl_matrix_submatrix

but can be used for matrices which are declared const.

[Function] gsl_matrix_view gsl_matrix_view_array (double * base, size t n1,

size t n2)

[Function] gsl_matrix_const_view gsl_matrix_const_view_array (const

double * base, size t n1, size t n2) These functions return a matrix view of the array base. The matrix has n1 rows and n2 columns. The physical number of columns in memory is also given by n2. Mathematically, the (i, j)-th element of the new matrix is given by,

     m'(i,j) = base[i*n2 + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1. The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_array is equivalent to gsl_matrix_view_

array but can be used for matrices which are declared const.

[Function] gsl_matrix_view gsl_matrix_view_array_with_tda (double *

base, size t n1, size t n2, size t tda)

[Function] gsl_matrix_const_view gsl_matrix_const_view_array_with_tda

(const double * base, size t n1, size t n2, size t tda) These functions return a matrix view of the array base with a physical number of columns tda which may differ from the corresponding dimension of the matrix. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i, j)-th element of the new matrix is given by,

     m'(i,j) = base[i*tda + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1. The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_array_with_tda is equivalent to gsl_

matrix_view_array_with_tda but can be used for matrices which are declared

const.

[Function] gsl_matrix_view gsl_matrix_view_vector (gsl vector * v, size t n1,

size t n2)

[Function] gsl_matrix_const_view gsl_matrix_const_view_vector (const

gsl vector * v, size t n1, size t n2) These functions return a matrix view of the vector v. The matrix has n1 rows and n2 columns. The vector must have unit stride. The physical number of columns in memory is also given by n2. Mathematically, the (i, j)-th element of the new matrix is given by,

     m'(i,j) = v->data[i*n2 + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1.

The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_vector is equivalent to gsl_matrix_view_

vector but can be used for matrices which are declared const.

[Function] gsl_matrix_view gsl_matrix_view_vector_with_tda (gsl vector *

v, size t n1, size t n2, size t tda)

[Function] gsl_matrix_const_view
   gsl_matrix_const_view_vector_with_tda (const gsl vector * v, size t

n1, size t n2, size t tda) These functions return a matrix view of the vector v with a physical number of columns tda which may differ from the corresponding matrix dimension. The vector must have unit stride. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i, j)-th element of the new matrix is given by,

     m'(i,j) = v->data[i*tda + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1.

The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_vector_with_tda is equivalent to
gsl_matrix_view_vector_with_tda but can be used for matrices which are

declared const.

8.4.6 Creating row and column views

In general there are two ways to access an object, by reference or by copying. The functions described in this section create vector views which allow access to a row or column of a matrix by reference. Modifying elements of the view is equivalent to modifying the matrix, since both the vector view and the matrix point to the same memory block.

[Function] gsl_vector_view gsl_matrix_row (gsl matrix * m, size t i)

[Function] gsl_vector_const_view gsl_matrix_const_row (const gsl matrix *

m, size t i) These functions return a vector view of the i-th row of the matrix m. The data pointer of the new vector is set to null if i is out of range.

The function gsl_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.

8.4.7 Copying matrices

[Function] int gsl_matrix_memcpy (gsl matrix * dest, const gsl matrix * src)

This function copies the elements of the matrix src into the matrix dest. The two matrices must have the same size.

[Function] int gsl_matrix_swap (gsl matrix * m1, gsl matrix * m2)

This function exchanges the elements of the matrices m1 and m2 by copying. The two matrices must have the same size.

8.4.8 Copying rows and columns

The functions described in this section copy a row or column of a matrix into a vector. This allows the elements of the vector and the matrix to be modified independently. Note that if the matrix and the vector point to overlapping regions of memory then the result will be undefined. The same effect can be achieved with more generality using gsl_vector_memcpy with vector views of rows and columns.

[Function] int gsl_matrix_get_row (gsl vector * v, const gsl matrix * m, size t i)

This function copies the elements of the i-th row of the matrix m into the vector v. The length of the vector must be the same as the length of the row.

[Function] int gsl_matrix_get_col (gsl vector * v, const gsl matrix * m, size t j)

This function copies the elements of the j-th column of the matrix m into the vector v. The length of the vector must be the same as the length of the column.

[Function] int gsl_matrix_set_row (gsl matrix * m, size t i, const gsl vector * v)

This function copies the elements of the vector v into the i-th row of the matrix m. The length of the vector must be the same as the length of the row.

[Function] int gsl_matrix_set_col (gsl matrix * m, size t j, const gsl vector * v)

This function copies the elements of the vector v into the j-th column of the matrix m. The length of the vector must be the same as the length of the column.

8.4.9 Exchanging rows and columns

The following functions can be used to exchange the rows and columns of a matrix.

[Function] int gsl_matrix_swap_rows (gsl matrix * m, size t i, size t j)

This function exchanges the i-th and j-th rows of the matrix m in-place.

[Function] int gsl_matrix_swap_columns (gsl matrix * m, size t i, size t j)

This function exchanges the i-th and j-th columns of the matrix m in-place.

[Function] int gsl_matrix_swap_rowcol (gsl matrix * m, size t i, size t j)

This function exchanges the i-th row and j-th column of the matrix m in-place. The matrix must be square for this operation to be possible.

[Function] int gsl_matrix_transpose_memcpy (gsl matrix * dest, const

gsl matrix * src) This function makes the matrix dest the transpose of the matrix src by copying the elements of src into dest. This function works for all matrices provided that the dimensions of the matrix dest match the transposed dimensions of the matrix src.

[Function] int gsl_matrix_transpose (gsl matrix * m)

This function replaces the matrix m by its transpose by copying the elements of the matrix in-place. The matrix must be square for this operation to be possible.

8.4.10 Matrix operations

The following operations are defined for real and complex matrices.

[Function] int gsl_matrix_add (gsl matrix * a, const gsl matrix * b)

This function adds the elements of matrix b to the elements of matrix a, 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) = xa(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.

8.4.11 Finding maximum and minimum elements of matrices

The following operations are only defined for real matrices.

[Function] double gsl_matrix_max (const gsl matrix * m)

This function returns the maximum value in the matrix m.

[Function] double gsl_matrix_min (const gsl matrix * m)

This function returns the minimum value in the matrix m.

[Function] void gsl_matrix_minmax (const gsl matrix * m, double * min_out,

double * max_out) This function returns the minimum and maximum values in the matrix m, storing them in min out and max out.

[Function] void gsl_matrix_max_index (const gsl matrix * m, size t * imax, size t

   * jmax)

This function returns the indices of the maximum value in the matrix m, storing them in imax and jmax. When there are several equal maximum elements then the first element found is returned, searching in row-major order.

[Function] void gsl_matrix_min_index (const gsl matrix * m, size t * imin, size t

   * jmin)

This function returns the indices of the minimum value in the matrix m, storing them in imin and jmin. When there are several equal minimum elements then the first element found is returned, searching in row-major order.

[Function] void gsl_matrix_minmax_index (const gsl matrix * m, size t * imin,

size t * jmin, size t * imax, size t * jmax) This function returns the indices of the minimum and maximum values in the matrix m, storing them in (imin,jmin) and (imax,jmax). When there are several equal minimum or maximum elements then the first elements found are returned, searching in row-major order.

8.4.12 Matrix properties

[Function] int gsl_matrix_isnull (const gsl matrix * m)

This function returns 1 if all the elements of the matrix m are zero, and 0 otherwise.

8.4.13 Example programs for matrices

The program below shows how to allocate, initialize and read from a matrix using the

functions gsl_matrix_alloc, gsl_matrix_set and gsl_matrix_get.
#include <stdio.h>
#include <gsl/gsl_matrix.h>
int
main (void)
{
  int i, j;
  gsl_matrix * m = gsl_matrix_alloc (10, 3);
  for (i = 0; i < 10; i++)
    for (j = 0; j < 3; j++)
      gsl_matrix_set (m, i, j, 0.23 + 100*i + j);
  for (i = 0; i < 100; i++)
    for (j = 0; j < 3; j++)
      printf ("m(%d,%d) = %g\n", i, j,
              gsl_matrix_get (m, i, j));
  return 0;
}

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

$ ./a.out
m(0,0) = 0.23
m(0,1) = 1.23
m(0,2) = 2.23
m(1,0) = 100.23
m(1,1) = 101.23
m(1,2) = 102.23
...
m(9,2) = 902.23

gsl: matrix_source.c:13: ERROR: first index out of range
Default GSL error handler invoked.
Aborted (core dumped)

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

#include <stdio.h>
#include <gsl/gsl_matrix.h>
int
main (void)
{
int i, j, k = 0;
gsl_matrix * m = gsl_matrix_alloc (100, 100);
gsl_matrix * a = gsl_matrix_alloc (100, 100);
for (i = 0; i < 100; i++)
  for (j = 0; j < 100; j++)
    gsl_matrix_set (m, i, j, 0.23 + i + j);
{
   FILE * f = fopen ("test.dat", "wb");
   gsl_matrix_fwrite (f, m);
   fclose (f);
}
{
   FILE * f = fopen ("test.dat", "rb");
   gsl_matrix_fread (f, a);
   fclose (f);
}
for (i = 0; i < 100; i++)
  for (j = 0; j < 100; j++)
    {
      double mij = gsl_matrix_get (m, i, j);
      double aij = gsl_matrix_get (a, i, j);
      if (mij != aij) k++;
    }
printf ("differences = %d (should be zero)\n", k);
return (k > 0);
}

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

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

#include <math.h>
#include <stdio.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
int
main (void)
{
  size_t i,j;
  gsl_matrix *m = gsl_matrix_alloc (10, 10);
  for (i = 0; i < 10; i++)
    for (j = 0; j < 10; j++)
      gsl_matrix_set (m, i, j, sin (i) + cos (j));
  for (j = 0; j < 10; j++)
    {
      gsl_vector_view column = gsl_matrix_column (m, j);
      double d;
      d = gsl_blas_dnrm2 (&column.vector);
      printf ("matrix column %d, norm = %g\n", j, d);
    }
  gsl_matrix_free (m);
  return 0;
}

Here is the output of the program,

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

The results can be confirmed using gnu octave,

$ octave
GNU Octave, version 2.0.16.92
octave> m = sin(0:9)' * ones(1,10)

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

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.

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′ = vP rather than column-vectors, when permuting the elements of a vector.

The functions described in this chapter are defined in the header file

‘gsl_permutation.h’.

9.1 The Permutation struct

A permutation is defined by a structure containing two components, the size of the permutation and a pointer to the permutation array. The elements of the permutation array are all of type size_t. The gsl_permutation structure looks like this,

typedef struct
{
  size_t size;
  size_t * data;
} gsl_permutation;

9.2 Permutation allocation

[Function] gsl_permutation * gsl_permutation_alloc (size t n)

This function allocates memory for a new permutation of size n. The permutation is not initialized and its elements are undefined. Use the function gsl_permutation_ calloc if you want to create a permutation which is initialized to the identity. A null pointer is returned if insufficient memory is available to create the permutation.

[Function] gsl_permutation * gsl_permutation_calloc (size t n)

This function allocates memory for a new permutation of size n and initializes it to the identity. A null pointer is returned if insufficient memory is available to create the permutation.

[Function] void gsl_permutation_init (gsl permutation * p)

This function initializes the permutation p to the identity, i.e. (0, 1, 2, …, n − 1).

[Function] void gsl_permutation_free (gsl permutation * p)

This function frees all the memory used by the permutation p.

[Function] int gsl_permutation_memcpy (gsl permutation * dest, const

gsl permutation * src) This function copies the elements of the permutation src into the permutation dest. The two permutations must have the same size.

9.3 Accessing permutation elements

The following functions can be used to access and manipulate permutations.

[Function] size_t gsl_permutation_get (const gsl permutation * p, const size t

   i)

This function returns the value of the i-th element of the permutation p. If i lies outside the allowed range of 0 to n − 1 then the error handler is invoked and 0 is returned.

[Function] int gsl_permutation_swap (gsl permutation * p, const size t i, const

size t j) This function exchanges the i-th and j-th elements of the permutation p.

9.4 Permutation properties

[Function] size_t gsl_permutation_size (const gsl permutation * p)

This function returns the size of the permutation p.

[Function] size_t * gsl_permutation_data (const gsl permutation * p)

This function returns a pointer to the array of elements in the permutation p.

[Function] int gsl_permutation_valid (gsl permutation * p)

This function checks that the permutation p is valid. The n elements should contain each of the numbers 0 to n − 1 once and only once.

9.5 Permutation functions

[Function] void gsl_permutation_reverse (gsl permutation * p)

This function reverses the elements of the permutation p.

[Function] int gsl_permutation_inverse (gsl permutation * inv, const

gsl permutation * p) This function computes the inverse of the permutation p, storing the result in inv.

[Function] int gsl_permutation_next (gsl permutation * p)

This function advances the permutation p to the next permutation in lexicographic order and returns GSL_SUCCESS. If no further permutations are available it returns GSL_FAILURE and leaves p unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order.

[Function] int gsl_permutation_prev (gsl permutation * p)

This function steps backwards from the permutation p to the previous permutation in lexicographic order, returning GSL_SUCCESS. If no previous permutation is available it returns GSL_FAILURE and leaves p unmodified.

9.6 Applying Permutations

[Function] int gsl_permute (const size t * p, double * data, size t stride, size t

   n)

This function applies the permutation p to the array data of size n with stride stride.

[Function] int gsl_permute_inverse (const size t * p, double * data, size t

   stride, size t n)

This function applies the inverse of the permutation p to the array data of size n with stride stride.

[Function] int gsl_permute_vector (const gsl permutation * p, gsl vector * v)

This function applies the permutation p to the elements of the vector v, considered as a row-vector acted on by a permutation matrix from the right, v′ = vP. 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′ = vP^{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.

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.

9.8 Permutations in cyclic form

A permutation can be represented in both linear and cyclic notations. The functions described in this section convert between the two forms. The linear notation is an index mapping, and has already been described above. The cyclic notation expresses a permutation as a series of circular rearrangements of groups of elements, or cycles.

For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different sets of elements can be combined independently, for example (1 2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange of elements 4 and 5. A cycle of length one represents an element which is unchanged by the permutation and is referred to as a singleton.

It can be shown that every permutation can be decomposed into combinations of cycles. The decomposition is not unique, but can always be rearranged into a standard canonical form by a reordering of elements. The library uses the canonical form defined in Knuth’s Art of Computer Programming (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.

The procedure for obtaining the canonical form given by Knuth is,

1. Write all singleton cycles explicitly

2. Within each cycle, put the smallest number first

3. Order the cycles in decreasing order of the first number in the cycle.

For example, the linear representation (2 4 3 0 1) is represented as (1 4) (0 2 3) in canonical form. The permutation corresponds to an exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.

The important property of the canonical form is that it can be reconstructed from the contents of each cycle without the brackets. In addition, by removing the brackets it can be considered as a linear representation of a different permutation. In the example given above the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has many applications in the theory of permutations.

[Function] int gsl_permutation_linear_to_canonical (gsl permutation * q,

const gsl permutation * p) This function computes the canonical form of the permutation p and stores it in the output argument q.

[Function] int gsl_permutation_canonical_to_linear (gsl permutation * p,

const gsl permutation * q) This function converts a permutation q in canonical form back into linear form storing it in the output argument p.

[Function] size_t gsl_permutation_inversions (const gsl permutation * p)

This function counts the number of inversions in the permutation p. An inversion is any pair of elements that are not in order. For example, the permutation 2031 has three inversions, corresponding to the pairs (2,0) (2,1) and (3,1). The identity permutation has no inversions.

[Function] size_t gsl_permutation_linear_cycles (const gsl permutation * p)

This function counts the number of cycles in the permutation p, given in linear form.

[Function] size_t gsl_permutation_canonical_cycles (const gsl permutation

* q) This function counts the number of cycles in the permutation q, given in canonical form.

9.9 Examples

The example program below creates a random permutation (by shuffling the elements of the identity) and finds its inverse.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_permutation.h>
int
main (void)
{
  const size_t N = 10;
  const gsl_rng_type * T;
  gsl_rng * r;
  gsl_permutation * p = gsl_permutation_alloc (N);
  gsl_permutation * q = gsl_permutation_alloc (N);
  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);
  printf ("initial permutation:");
  gsl_permutation_init (p);
  gsl_permutation_fprintf (stdout, p, " %u");
  printf ("\n");
  printf (" random permutation:");

gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
gsl_permutation_fprintf (stdout, p, " %u");
printf ("\n");
printf ("inverse permutation:");
gsl_permutation_inverse (q, p);
gsl_permutation_fprintf (stdout, q, " %u");
printf ("\n");
return 0;
}

Here is the output from the program,

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

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

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

#include <stdio.h>
#include <gsl/gsl_permutation.h>
int
main (void)
{
gsl_permutation * p = gsl_permutation_alloc (3);
gsl_permutation_init (p);
do
 {
    gsl_permutation_fprintf (stdout, p, " %u");
    printf ("\n");
 }
while (gsl_permutation_next(p) == GSL_SUCCESS);
return 0;
}

Here is the output from the program,

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

      2 0 1
      2 1 0

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

next with gsl_permutation_prev.

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.

10 Combinations

This chapter describes functions for creating and manipulating combinations. A combination c is represented by an array of k integers in the range 0 to n − 1, where each value c_{i} occurs at most once. The combination c corresponds to indices of k elements chosen from an n element vector. Combinations are useful for iterating over all k-element subsets of a set.

The functions described in this chapter are defined in the header file

‘gsl_combination.h’.

10.1 The Combination struct

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

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

10.2 Combination allocation

[Function] gsl_combination * gsl_combination_alloc (size t n, size t k)

This function allocates memory for a new combination with parameters n, k. The combination is not initialized and its elements are undefined. Use the function gsl_ combination_calloc if you want to create a combination which is initialized to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.

[Function] gsl_combination * gsl_combination_calloc (size t n, size t k)

This function allocates memory for a new combination with parameters n, k and initializes it to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.

[Function] void gsl_combination_init_first (gsl combination * c)

This function initializes the combination c to the lexicographically first combination, i.e. (0, 1, 2, …, 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.

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.

10.4 Combination properties

[Function] size_t gsl_combination_n (const gsl combination * c)

This function returns the range (n) of the combination c.

[Function] size_t gsl_combination_k (const gsl combination * c)

This function returns the number of elements (k) in the combination c.

[Function] size_t * gsl_combination_data (const gsl combination * c)

This function returns a pointer to the array of elements in the combination c.

[Function] int gsl_combination_valid (gsl combination * c)

This function checks that the combination c is valid. The k elements should lie in the range 0 to n − 1, with each value occurring once at most and in increasing order.

10.5 Combination functions

[Function] int gsl_combination_next (gsl combination * c)

This function advances the combination c to the next combination in lexicographic order and returns GSL_SUCCESS. If no further combinations are available it returns GSL_FAILURE and leaves c unmodified. Starting with the first combination and repeatedly applying this function will iterate through all possible combinations of a given order.

[Function] int gsl_combination_prev (gsl combination * c)

This function steps backwards from the combination c to the previous combination in lexicographic order, returning GSL_SUCCESS. If no previous combination is available it returns GSL_FAILURE and leaves c unmodified.

10.6 Reading and writing combinations

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

[Function] int gsl_combination_fwrite (FILE * stream, const gsl combination

* c) This function writes the elements of the combination c to the stream stream in binary format. The function returns GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

[Function] int gsl_combination_fread (FILE * stream, gsl combination * c)

This function reads elements from the open stream stream into the combination c in binary format. The combination c must be preallocated with correct values of n and k since the function uses the size of c to determine how many bytes to read. The function returns GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

[Function] int gsl_combination_fprintf (FILE * stream, const gsl combination

* c, const char * format) This function writes the elements of the combination c line-by-line to the stream stream using the format specifier format, which should be suitable for a type of size t. 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.

10.7 Examples

The example program below prints all subsets of the set {0, 1, 2, 3} ordered by size. Subsets of the same size are ordered lexicographically.

#include <stdio.h>
#include <gsl/gsl_combination.h>
int
main (void)
{
  gsl_combination * c;
  size_t i;
  printf ("All subsets of {0,1,2,3} by size:\n") ;
  for (i = 0; i <= 4; i++)
    {
      c = gsl_combination_calloc (4, i);
      do
        {
          printf ("{");

          gsl_combination_fprintf (stdout, c, " %u");
          printf (" }\n");
        }
      while (gsl_combination_next (c) == GSL_SUCCESS);
      gsl_combination_free (c);
    }
  return 0;
}

Here is the output from the program,

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

All 16 subsets are generated, and the subsets of each size are sorted lexicographically.

10.8 References and Further Reading

Further information on combinations can be found in, Donald L. Kreher, Douglas R. Stinson, Combinatorial Algorithms: Generation, Enumeration and Search, 1998, CRC Press LLC, ISBN 084933988X

11 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.

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.

11.2 Sorting vectors

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

header files ‘gsl_sort.h’ and ‘gsl_sort_vector.h’.

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

[Function] void gsl_sort (double * data, size t stride, size t n)

This function sorts the n elements of the array data with stride stride into ascending numerical order.

[Function] void gsl_sort_vector (gsl vector * v)

This function sorts the elements of the vector v into ascending numerical order.

[Function] int gsl_sort_index (size t * p, const double * data, size t stride,

size t n) This function indirectly sorts the n elements of the array data with stride stride into ascending order, storing the resulting permutation in p. The array p must be allocated with a sufficient length to store the n elements of the permutation. The elements of p give the index of the array element which would have been stored in that position if the array had been sorted in place. The array data is not changed.

[Function] int gsl_sort_vector_index (gsl permutation * p, const gsl vector *

   v)

This function indirectly sorts the elements of the vector v into ascending order, storing the resulting permutation in p. The elements of p give the index of the vector element which would have been stored in that position if the vector had been sorted in place.

The first element of p gives the index of the least element in v, and the last element of p gives the index of the greatest element in v. The vector v is not changed.

11.3 Selecting the k smallest or largest elements

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

[Function] void gsl_sort_smallest (double * dest, size t k, const double * src,

size t stride, size t n) This function copies the k smallest elements of the array src, of size n and stride stride, in ascending numerical order into the array dest. The size k of the subset must be less than or equal to n. The data src is not modified by this operation.

[Function] void gsl_sort_largest (double * dest, size t k, const double * src,

size t stride, size t n) This function copies the k largest elements of the array src, of size n and stride stride, in descending numerical order into the array dest. k must be less than or equal to n. The data src is not modified by this operation.

[Function] void gsl_sort_vector_smallest (double * dest, size t k, const

gsl vector * v)

[Function] void gsl_sort_vector_largest (double * dest, size t k, const

gsl vector * v) These functions copy the k smallest or largest elements of the vector v into the array dest. k must be less than or equal to the length of the vector v.

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

[Function] void gsl_sort_smallest_index (size t * p, size t k, const double *

src, size t stride, size t n) This function stores the indices of the k smallest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in ascending numerical order. k must be less than or equal to n. The data src is not modified by this operation.

[Function] void gsl_sort_largest_index (size t * p, size t k, const double *

src, size t stride, size t n) This function stores the indices of the k largest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in descending numerical order. k must be less than or equal to n. The data src is not modified by this operation.

[Function] void gsl_sort_vector_smallest_index (size t * p, size t k, const

gsl vector * v)

[Function] void gsl_sort_vector_largest_index (size t * p, size t k, const

gsl vector * v) These functions store the indices of the k smallest or largest elements of the vector v in the array p. k must be less than or equal to the length of the vector v.

11.4 Computing the rank

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

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

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

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

void
print_rank (gsl_vector * v)
{
  size_t i;
  size_t n = v->size;
  gsl_permutation * perm = gsl_permutation_alloc(n);
  gsl_permutation * rank = gsl_permutation_alloc(n);
  gsl_sort_vector_index (perm, v);
  gsl_permutation_inverse (rank, perm);
  for (i = 0; i < n; i++)
   {
    double vi = gsl_vector_get(v, i);
    printf ("element = %d, value = %g, rank = %d\n",
             i, vi, rank->data[i]);
   }
  gsl_permutation_free (perm);
  gsl_permutation_free (rank);
}

11.5 Examples

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

gsl_sort_vector_index (p, v);
for (i = 0; i < v->size; i++)
{
    double vpi = gsl_vector_get (v, p->data[i]);

    printf ("order = %d, value = %g\n", i, vpi);
}

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

#include <gsl/gsl_rng.h>
#include <gsl/gsl_sort_double.h>
int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;
  size_t i, k = 5, N = 100000;
  double * x = malloc (N * sizeof(double));
  double * small = malloc (k * sizeof(double));
  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);
  for (i = 0; i < N; i++)
    {
      x[i] = gsl_rng_uniform(r);
    }
  gsl_sort_smallest (small, k, x, 1, N);
  printf ("%d smallest values from %d\n", k, N);
  for (i = 0; i < k; i++)
    {
      printf ("%d: %.18f\n", i, small[i]);
    }
  return 0;
}

The output lists the 5 smallest values, in ascending order,

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

11.6 References and Further Reading

The subject of sorting is covered extensively in Knuth’s Sorting and Searching, Donald E. Knuth, The Art of Computer Programming: Sorting and Searching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

The Heapsort algorithm is described in the following book, Robert Sedgewick, Algorithms in C, Addison-Wesley, ISBN 0201514257.

12 BLAS Support

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

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

The interface for the gsl_cblas layer is specified in the file gsl_cblas.h. This interface corresponds to the blas Technical Forum’s draft standard for the C interface to legacy blas implementations. Users who have access to other conforming cblas implementations can use these in place of the version provided by the library. Note that users who have only a Fortran blas library can use a cblas conformant wrapper to convert it into a cblas library. A reference cblas wrapper for legacy Fortran implementations exists as part of the draft cblas standard and can be obtained from Netlib. The complete set of cblas functions is listed in an appendix (see Appendix D [GSL CBLAS Library], page 411).

There are three levels of blas operations,

Level 1 Vector operations, e.g. y = αx + y

Level 2 Matrix-vector operations, e.g. y = αAx + βy

Level 3 Matrix-matrix operations, e.g. C = αAB + C

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

DOT scalar product, x^{T} y

AXPY vector sum, αx + y

MV matrix-vector product, Ax

SV matrix-vector solve, inv(A)x

MM matrix-matrix product, AB

SM matrix-matrix solve, inv(A)B

The types of matrices are,

GE general

GB general band

SY symmetric

SB symmetric band

SP symmetric packed HE hermitian

HB hermitian band

HP hermitian packed

TR triangular

TB triangular band

TP triangular packed

Each operation is defined for four precisions,

S single real

D double real

C single complex

Z double complex

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

12.1 GSL BLAS Interface

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

12.1.1 Level 1

[Function] int gsl_blas_sdsdot (float alpha, const gsl vector float * x, const

gsl vector float * y, float * result) This function computes the sum α + x^{T} y for the vectors x and y, returning the result in result.

[Function] int gsl_blas_sdot (const gsl vector float * x, const gsl vector float * y,

        float * result)

[Function] int gsl_blas_dsdot (const gsl vector float * x, const gsl vector float * y, double * result)

[Function] int gsl_blas_ddot (const gsl vector * x, const gsl vector * y, double *

        result)

These functions compute the scalar product x^{T} y for the vectors x and y, returning the result in result.

[Function] int gsl_blas_cdotu (const gsl vector complex float * x, const gsl vector complex float * y, gsl complex float * dotu)

[Function] int gsl_blas_zdotu (const gsl vector complex * x, const

gsl vector complex * y, gsl complex * dotu) These functions compute the complex scalar product x^{T} y for the vectors x and y, returning the result in result [Function] int gsl_blas_cdotc (const gsl vector complex float * x, const

gsl vector complex float * y, gsl complex float * dotc)

[Function] int gsl_blas_zdotc (const gsl vector complex * x, const

gsl vector complex * y, gsl complex * dotc) These functions compute the complex conjugate scalar product x^{H}y for the vectors x and y, returning the result in result

[Function] float gsl_blas_snrm2 (const gsl vector float * x)

[Function] double gsl_blas_dnrm2 (const gsl vector * x)

These functions compute the Euclidean norm ||x||_{2} = \sqrt{∑ x_{i}^{2}} of the vector x.

[Function] float gsl_blas_scnrm2 (const gsl vector complex float * x)

[Function] double gsl_blas_dznrm2 (const gsl vector complex * x)

These functions compute the Euclidean norm of the complex vector x,

||x||_{2} = \sqrt{∑(Re(x_{i})^{2} + Im(x_{i})^{2})}.

[Function] float gsl_blas_sasum (const gsl vector float * x)

[Function] double gsl_blas_dasum (const gsl vector * x)

These functions compute the absolute sum ∑ |x_{i}| of the elements of the vector x.

[Function] float gsl_blas_scasum (const gsl vector complex float * x)

[Function] double gsl_blas_dzasum (const gsl vector complex * x)

These functions compute the sum of the magnitudes of the real and imaginary parts of the complex vector x, ∑ (|Re(x_{i})| + |Im(x_{i})|).

[Function] CBLAS_INDEX_t gsl_blas_isamax (const gsl vector float * x)

[Function] CBLAS_INDEX_t gsl_blas_idamax (const gsl vector * x)

[Function] CBLAS_INDEX_t gsl_blas_icamax (const gsl vector complex float * x)

[Function] CBLAS_INDEX_t gsl_blas_izamax (const gsl vector complex * x)

These functions return the index of the largest element of the vector x. The largest element is determined by its absolute magnitude for real vectors and by the sum of the magnitudes of the real and imaginary parts |Re(x_{i})| + |Im(x_{i})| for complex vectors. If the largest value occurs several times then the index of the first occurrence is returned.

[Function] int gsl_blas_sswap (gsl vector float * x, gsl vector float * y)

[Function] int gsl_blas_dswap (gsl vector * x, gsl vector * y)

[Function] int gsl_blas_cswap (gsl vector complex float * x, gsl vector complex float * y)

[Function] int gsl_blas_zswap (gsl vector complex * x, gsl vector complex * y)

These functions exchange the elements of the vectors x and y.

[Function] int gsl_blas_scopy (const gsl vector float * x, gsl vector float * y)

[Function] int gsl_blas_dcopy (const gsl vector * x, gsl vector * y)

[Function] int gsl_blas_ccopy (const gsl vector complex float * x, gsl vector complex float * y)

[Function] int gsl_blas_zcopy (const gsl vector complex * x, gsl vector complex

* y) These functions copy the elements of the vector x into the vector y.

[Function] int gsl_blas_saxpy (float alpha, const gsl vector float * x, gsl vector float * y)

[Function] int gsl_blas_daxpy (double alpha, const gsl vector * x, gsl vector * y)

[Function] int gsl_blas_caxpy (const gsl complex float alpha, const gsl vector complex float * x, gsl vector complex float * y)

[Function] int gsl_blas_zaxpy (const gsl complex alpha, const

gsl vector complex * x, gsl vector complex * y) These functions compute the sum y = αx + y for the vectors x and y.

[Function] void gsl_blas_sscal (float alpha, gsl vector float * x)

[Function] void gsl_blas_dscal (double alpha, gsl vector * x)

[Function] void gsl_blas_cscal (const gsl complex float alpha,

gsl vector complex float * x)

[Function] void gsl_blas_zscal (const gsl complex alpha, gsl vector complex *

        x)

[Function] void gsl_blas_csscal (float alpha, gsl vector complex float * x)

[Function] void gsl_blas_zdscal (double alpha, gsl vector complex * x)

These functions rescale the vector x by the multiplicative factor alpha.

[Function] int gsl_blas_srotg (float a[], float b[], float c[], float s[])

[Function] int gsl_blas_drotg (double a[], double b[], double c[], double s[])

These functions compute a Givens rotation (c, s) which zeroes the vector (a, b),

( \begin{matrix}c & s \\ −s & c \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) = ( \begin{matrix}r′ \\ 0 \end{matrix} )

The variables a and b are overwritten by the routine.

[Function] int gsl_blas_srot (gsl vector float * x, gsl vector float * y, float c, float s)

[Function] int gsl_blas_drot (gsl vector * x, gsl vector * y, const double c, const

double s) These functions apply a Givens rotation (x′, y′) = (cx + sy, −sx + cy) to the vectors x, y.

[Function] int gsl_blas_srotmg (float d1[], float d2[], float b1[], float b2, float P[])

[Function] int gsl_blas_drotmg (double d1[], double d2[], double b1[], double b2,

double P[]) These functions compute a modified Givens transformation. The modified Givens transformation is defined in the original Level-1 blas specification, given in the references.

[Function] int gsl_blas_srotm (gsl vector float * x, gsl vector float * y, const float P[])

[Function] int gsl_blas_drotm (gsl vector * x, gsl vector * y, const double P[])

These functions apply a modified Givens transformation.

12.1.2 Level 2

[Function] int gsl_blas_sgemv (CBLAS TRANSPOSE t TransA, float alpha,

const gsl matrix float * A, const gsl vector float * x, float beta, gsl vector float * y)

[Function] int gsl_blas_dgemv (CBLAS TRANSPOSE t TransA, double alpha,

const gsl matrix * A, const gsl vector * x, double beta, gsl vector * y)

[Function] int gsl_blas_cgemv (CBLAS TRANSPOSE t TransA, const

gsl complex float alpha, const gsl matrix complex float * A, const gsl vector complex float * x, const gsl complex float beta, gsl vector complex float * y)

[Function] int gsl_blas_zgemv (CBLAS TRANSPOSE t TransA, const

gsl complex alpha, const gsl matrix complex * A, const gsl vector complex * x, const gsl complex beta, gsl vector complex * y)

These functions compute the matrix-vector product and sum y = αop(A)x+βy, where

op(A) = A, A^{T}, A^{H} for TransA = CblasNoTrans, CblasTrans, CblasConjTrans.

[Function] int gsl_blas_strmv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix float * A, gsl vector float * x)

[Function] int gsl_blas_dtrmv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix * A, gsl vector * x)

[Function] int gsl_blas_ctrmv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix complex float * A, gsl vector complex float * x)

[Function] int gsl_blas_ztrmv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix complex * A, gsl vector complex * x)

These functions compute the matrix-vector product x = αop(A)x for the triangular matrix A, where op(A) = A, A^{T}, A^{H} for TransA = CblasNoTrans, CblasTrans, CblasConjTrans. When Uplo is CblasUpper then the upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of the matrix is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

[Function] int gsl_blas_strsv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix float * A, gsl vector float * x)

[Function] int gsl_blas_dtrsv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix * A, gsl vector * x)

[Function] int gsl_blas_ctrsv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix complex float * A, gsl vector complex float * x)

[Function] int gsl_blas_ztrsv (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl matrix complex * A, gsl vector complex * x)

These functions compute inv(op(A))x for x, where op(A) = A, A^{T}, A^{H} for TransA =

CblasNoTrans, CblasTrans, CblasConjTrans. When Uplo is CblasUpper then the

upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of the matrix is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

[Function] int gsl_blas_ssymv (CBLAS UPLO t Uplo, float alpha, const

gsl matrix float * A, const gsl vector float * x, float beta, gsl vector float * y)

[Function] int gsl_blas_dsymv (CBLAS UPLO t Uplo, double alpha, const

gsl matrix * A, const gsl vector * x, double beta, gsl vector * y) These functions compute the matrix-vector product and sum y = αAx + βy for the symmetric matrix A. Since the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

[Function] int gsl_blas_chemv (CBLAS UPLO t Uplo, const gsl complex float alpha, const gsl matrix complex float * A, const gsl vector complex float * x, const gsl complex float beta, gsl vector complex float * y)

[Function] int gsl_blas_zhemv (CBLAS UPLO t Uplo, const gsl complex alpha, const gsl matrix complex * A, const gsl vector complex * x, const gsl complex beta, gsl vector complex * y)

These functions compute the matrix-vector product and sum y = αAx + βy for the hermitian matrix A. Since the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically assumed to be zero and are not referenced.

[Function] int gsl_blas_sger (float alpha, const gsl vector float * x, const gsl vector float * y, gsl matrix float * A)

[Function] int gsl_blas_dger (double alpha, const gsl vector * x, const gsl vector * y, gsl matrix * A)

[Function] int gsl_blas_cgeru (const gsl complex float alpha, const gsl vector complex float * x, const gsl vector complex float * y, gsl matrix complex float * A)

[Function] int gsl_blas_zgeru (const gsl complex alpha, const

gsl vector complex * x, const gsl vector complex * y, gsl matrix complex * A) These functions compute the rank-1 update A = αxy^{T} + A of the matrix A.

[Function] int gsl_blas_cgerc (const gsl complex float alpha, const gsl vector complex float * x, const gsl vector complex float * y, gsl matrix complex float * A)

[Function] int gsl_blas_zgerc (const gsl complex alpha, const

gsl vector complex * x, const gsl vector complex * y, gsl matrix complex * A) These functions compute the conjugate rank-1 update A = αxy^{H} + A of the matrix A.

[Function] int gsl_blas_ssyr (CBLAS UPLO t Uplo, float alpha, const

gsl vector float * x, gsl matrix float * A)

[Function] int gsl_blas_dsyr (CBLAS UPLO t Uplo, double alpha, const

gsl vector * x, gsl matrix * A) These functions compute the symmetric rank-1 update A = αxx^{T} +A of the symmetric matrix A. Since the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

[Function] int gsl_blas_cher (CBLAS UPLO t Uplo, float alpha, const

gsl vector complex float * x, gsl matrix complex float * A)

[Function] int gsl_blas_zher (CBLAS UPLO t Uplo, double alpha, const

gsl vector complex * x, gsl matrix complex * A) These functions compute the hermitian rank-1 update A = αxx^{H} +A of the hermitian matrix A. Since the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.

[Function] int gsl_blas_ssyr2 (CBLAS UPLO t Uplo, float alpha, const

gsl vector float * x, const gsl vector float * y, gsl matrix float * A)

[Function] int gsl_blas_dsyr2 (CBLAS UPLO t Uplo, double alpha, const

gsl vector * x, const gsl vector * y, gsl matrix * A) These functions compute the symmetric rank-2 update A = αxy^{T} + αyx^{T} + A of the symmetric matrix A. Since the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

[Function] int gsl_blas_cher2 (CBLAS UPLO t Uplo, const gsl complex float alpha, const gsl vector complex float * x, const gsl vector complex float * y, gsl matrix complex float * A)

[Function] int gsl_blas_zher2 (CBLAS UPLO t Uplo, const gsl complex alpha, const gsl vector complex * x, const gsl vector complex * y, gsl matrix complex * A)

These functions compute the hermitian rank-2 update A = αxy^{H} + α^{∗}yx^{H}A of the hermitian matrix A. Since the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.

12.1.3 Level 3

[Function] int gsl_blas_sgemm (CBLAS TRANSPOSE t TransA,

CBLAS TRANSPOSE t TransB, float alpha, const gsl matrix float * A, const gsl matrix float * B, float beta, gsl matrix float * C)

[Function] int gsl_blas_dgemm (CBLAS TRANSPOSE t TransA,

CBLAS TRANSPOSE t TransB, double alpha, const gsl matrix * A, const gsl matrix * B, double beta, gsl matrix * C)

[Function] int gsl_blas_cgemm (CBLAS TRANSPOSE t TransA,

CBLAS TRANSPOSE t TransB, const gsl complex float alpha, const gsl matrix complex float * A, const gsl matrix complex float * B, const gsl complex float beta, gsl matrix complex float * C)

[Function] int gsl_blas_zgemm (CBLAS TRANSPOSE t TransA,

CBLAS TRANSPOSE t TransB, const gsl complex alpha, const gsl matrix complex * A, const gsl matrix complex * B, const gsl complex beta, gsl matrix complex * C)

These functions compute the matrix-matrix product and sum C = αop(A)op(B)+βC where op(A) = A, A^{T}, A^{H} for TransA = CblasNoTrans, CblasTrans, CblasConjTrans and similarly for the parameter TransB.

[Function] int gsl_blas_ssymm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

float alpha, const gsl matrix float * A, const gsl matrix float * B, float beta, gsl matrix float * C)

[Function] int gsl_blas_dsymm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

double alpha, const gsl matrix * A, const gsl matrix * B, double beta, gsl matrix * C)

[Function] int gsl_blas_csymm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

const gsl complex float alpha, const gsl matrix complex float * A, const gsl matrix complex float * B, const gsl complex float beta, gsl matrix complex float * C)

[Function] int gsl_blas_zsymm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

const gsl complex alpha, const gsl matrix complex * A, const gsl matrix complex * B, const gsl complex beta, gsl matrix complex * C)

These functions compute the matrix-matrix product and sum C = αAB + βC for Side is CblasLeft and C = αBA + βC for Side is CblasRight, where the matrix A is symmetric. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

[Function] int gsl_blas_chemm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

const gsl complex float alpha, const gsl matrix complex float * A, const gsl matrix complex float * B, const gsl complex float beta, gsl matrix complex float * C)

[Function] int gsl_blas_zhemm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

const gsl complex alpha, const gsl matrix complex * A, const gsl matrix complex * B, const gsl complex beta, gsl matrix complex * C)

These functions compute the matrix-matrix product and sum C = αAB + βC for Side is CblasLeft and C = αBA + βC for Side is CblasRight, where the matrix A is hermitian. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.

[Function] int gsl_blas_strmm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, float alpha, const gsl matrix float * A, gsl matrix float * B)

[Function] int gsl_blas_dtrmm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, double alpha, const gsl matrix * A, gsl matrix * B)

[Function] int gsl_blas_ctrmm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl complex float alpha, const gsl matrix complex float * A, gsl matrix complex float * B)

[Function] int gsl_blas_ztrmm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl complex alpha, const gsl matrix complex * A, gsl matrix complex * B)

These functions compute the matrix-matrix product B = αop(A)B for Side is CblasLeft and B = αBop(A) for Side is CblasRight. The matrix A is triangular

and op(A) = A, A^{T}, A^{H} for TransA = CblasNoTrans, CblasTrans, CblasConjTrans

When Uplo is CblasUpper then the upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of A is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

[Function] int gsl_blas_strsm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, float alpha, const gsl matrix float * A, gsl matrix float * B)

[Function] int gsl_blas_dtrsm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, double alpha, const gsl matrix * A, gsl matrix * B)

[Function] int gsl_blas_ctrsm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl complex float alpha, const gsl matrix complex float * A, gsl matrix complex float * B)

[Function] int gsl_blas_ztrsm (CBLAS SIDE t Side, CBLAS UPLO t Uplo,

CBLAS TRANSPOSE t TransA, CBLAS DIAG t Diag, const gsl complex alpha, const gsl matrix complex * A, gsl matrix complex * B)

These functions compute the inverse-matrix matrix product B = αop(inv(A))B for Side is CblasLeft and B = αBop(inv(A)) for Side is CblasRight. The matrix A is triangular and op(A) = A, A^{T}, A^{H} for TransA = CblasNoTrans, CblasTrans, CblasConjTrans When Uplo is CblasUpper then the upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of A is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

[Function] int gsl_blas_ssyrk (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, float alpha, const gsl matrix float * A, float beta, gsl matrix float *

C)

[Function] int gsl_blas_dsyrk (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, double alpha, const gsl matrix * A, double beta, gsl matrix * C)

[Function] int gsl_blas_csyrk (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, const gsl complex float alpha, const gsl matrix complex float * A, const gsl complex float beta, gsl matrix complex float * C)

[Function] int gsl_blas_zsyrk (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, const gsl complex alpha, const gsl matrix complex * A, const gsl complex beta, gsl matrix complex * C)

These functions compute a rank-k update of the symmetric matrix C, C = αAA^{T} +βC when Trans is CblasNoTrans and C = αA^{T} A + βC when Trans is CblasTrans. Since the matrix C is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used.

[Function] int gsl_blas_cherk (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, float alpha, const gsl matrix complex float * A, float beta, gsl matrix complex float * C)

[Function] int gsl_blas_zherk (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, double alpha, const gsl matrix complex * A, double beta, gsl matrix complex * C)

These functions compute a rank-k update of the hermitian matrix C, C = αAA^{H} +βC when Trans is CblasNoTrans and C = αA^{H}A+βC when Trans is CblasTrans. Since the matrix C is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used. The imaginary elements of the diagonal are automatically set to zero.

[Function] int gsl_blas_ssyr2k (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, float alpha, const gsl matrix float * A, const gsl matrix float * B, float beta, gsl matrix float * C)

[Function] int gsl_blas_dsyr2k (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, double alpha, const gsl matrix * A, const gsl matrix * B, double beta, gsl matrix * C)

[Function] int gsl_blas_csyr2k (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, const gsl complex float alpha, const gsl matrix complex float * A, const gsl matrix complex float * B, const gsl complex float beta, gsl matrix complex float * C)

[Function] int gsl_blas_zsyr2k (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, const gsl complex alpha, const gsl matrix complex * A, const gsl matrix complex * B, const gsl complex beta, gsl matrix complex * C)

These functions compute a rank-2k update of the symmetric matrix C, C = αAB^{T} + αBA^{T} + βC when Trans is CblasNoTrans and C = αA^{T} B + αB^{T} A + βC when Trans is CblasTrans. Since the matrix C is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used.

[Function] int gsl_blas_cher2k (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, const gsl complex float alpha, const gsl matrix complex float * A, const gsl matrix complex float * B, float beta, gsl matrix complex float * C)

[Function] int gsl_blas_zher2k (CBLAS UPLO t Uplo, CBLAS TRANSPOSE t Trans, const gsl complex alpha, const gsl matrix complex * A, const gsl matrix complex * B, double beta, gsl matrix complex * C)

These functions compute a rank-2k update of the hermitian matrix C, C = αAB^{H} + α^{∗}BA^{H} + βC when Trans is CblasNoTrans and C = αA^{H}B + α^{∗}B^{H}A + βC when Trans is CblasConjTrans. Since the matrix C is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used. The imaginary elements of the diagonal are automatically set to zero.

12.2 Examples

The following program computes the product of two matrices using the Level-3 blas function dgemm,

( \begin{matrix}0.11 & 0.12 & 0.13 \\ 0.21 & 0.22 & 0.23 \end{matrix} ) ( \begin{matrix}1011 & 1012 \\ 1021 & 1022 \\ 1031 & 1031 \end{matrix} ) = ( \begin{matrix}367.76 & 368.12 \\ 674.06 & 674.72 \end{matrix} )

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

#include <stdio.h>
#include <gsl/gsl_blas.h>
int
main (void)
{
  double a[] = { 0.11, 0.12, 0.13,
                 0.21, 0.22, 0.23 };
  double b[] = { 1011, 1012,
                 1021, 1022,
                 1031, 1032 };
  double c[] = { 0.00, 0.00,
                 0.00, 0.00 };
  gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
  gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
  gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);
  /* Compute C = A B */
  gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
                  1.0, &A.matrix, &B.matrix,
                  0.0, &C.matrix);
  printf ("[ %g, %g\n", c[0], c[1]);

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

Here is the output from the program,

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

12.3 References and Further Reading

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

BLAS Homepage

http://www.netlib.org/blas/

BLAS Technical Forum

http://www.netlib.org/cgi-bin/checkout/blast/blast.pl

The following papers contain the specifications for Level 1, Level 2 and Level 3 blas. C. Lawson, R. Hanson, D. Kincaid, F. Krogh, “Basic Linear Algebra Subprograms for Fortran Usage”, ACM Transactions on Mathematical Software, Vol. 5 (1979), Pages 308–325. J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, “An Extended Set of Fortran Basic Linear Algebra Subprograms”, ACM Transactions on Mathematical Software, Vol. 14, No. 1 (1988), Pages 1–32. J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, “A Set of Level 3 Basic Linear Algebra Subprograms”, ACM Transactions on Mathematical Software, Vol. 16 (1990), Pages 1–28.

Postscript versions of the latter two papers are available from http://www.netlib.org/blas/. A cblas wrapper for Fortran blas libraries is available from the same location.

13 Linear Algebra

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

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

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

13.1 LU Decomposition

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

P A = LU

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

[Function] int gsl_linalg_LU_decomp (gsl matrix * A, gsl permutation * p, int *

        signum)
[Function] int gsl_linalg_complex_LU_decomp (gsl matrix complex * A,

gsl permutation * p, int * signum) These functions factorize the square matrix A into the LU decomposition P A = LU. On output the diagonal and upper triangular part of the input matrix A contain the matrix U. The lower triangular part of the input matrix (excluding the diagonal) contains L. The diagonal elements of L are unity, and are not stored. The permutation matrix P is encoded in the permutation p. The j-th column of the matrix P is given by the k-th column of the identity matrix, where k = p_{j} the j-th element of the permutation vector. The sign of the permutation is given by signum. It has the value (−1)^{n}, where n is the number of interchanges in the permutation. The algorithm used in the decomposition is Gaussian Elimination with partial pivoting (Golub & Van Loan, Matrix Computations, Algorithm 3.4.1).

[Function] int gsl_linalg_LU_solve (const gsl matrix * LU, const

gsl permutation * p, const gsl vector * b, gsl vector * x)

[Function] int gsl_linalg_complex_LU_solve (const gsl matrix complex * LU,

const gsl permutation * p, const gsl vector complex * b, gsl vector complex *

        x)

These functions solve the square system Ax = b using the LU decomposition of A

     into (LU, p) given by gsl_linalg_LU_decomp or gsl_linalg_complex_LU_decomp.

[Function] int gsl_linalg_LU_svx (const gsl matrix * LU, const gsl permutation * p, gsl vector * x)

[Function] int gsl_linalg_complex_LU_svx (const gsl matrix complex * LU,

const gsl permutation * p, gsl vector complex * x) These functions solve the square system Ax = b in-place using the LU decomposition of A into (LU,p). On input x should contain the right-hand side b, which is replaced by the solution on output.

[Function] int gsl_linalg_LU_refine (const gsl matrix * A, const gsl matrix * LU, const gsl permutation * p, const gsl vector * b, gsl vector * x, gsl vector *

   residual)

[Function] int gsl_linalg_complex_LU_refine (const gsl matrix complex * A,

const gsl matrix complex * LU, const gsl permutation * p, const gsl vector complex * b, gsl vector complex * x, gsl vector complex *

   residual)

These functions apply an iterative improvement to x, the solution of Ax = b, using the LU decomposition of A into (LU,p). The initial residual r = Ax − b is also computed and stored in residual.

[Function] int gsl_linalg_LU_invert (const gsl matrix * LU, const

gsl permutation * p, gsl matrix * inverse)

[Function] int gsl_linalg_complex_LU_invert (const gsl matrix complex * LU,

const gsl permutation * p, gsl matrix complex * inverse) These functions compute the inverse of a matrix A from its LU decomposition (LU,p), storing the result in the matrix inverse. The inverse is computed by solving the system Ax = b for each column of the identity matrix. It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtain the same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for details).

[Function] double gsl_linalg_LU_det (gsl matrix * LU, int signum)
[Function] gsl_complex gsl_linalg_complex_LU_det (gsl matrix complex * LU,
   int signum)

These functions compute the determinant of a matrix A from its LU decomposition, LU. The determinant is computed as the product of the diagonal elements of U and the sign of the row permutation signum.

[Function] double gsl_linalg_LU_lndet (gsl matrix * LU)
[Function] double gsl_linalg_complex_LU_lndet (gsl matrix complex * LU)

These functions compute the logarithm of the absolute value of the determinant of a matrix A, ln |det(A)|, from its LU decomposition, LU. This function may be useful if the direct computation of the determinant would overflow or underflow.

[Function] int gsl_linalg_LU_sgndet (gsl matrix * LU, int signum)
[Function] gsl_complex gsl_linalg_complex_LU_sgndet (gsl matrix complex *
   LU, int signum)

These functions compute the sign or phase factor of the determinant of a matrix A, det(A)/|det(A)|, from its LU decomposition, LU.

13.2 QR Decomposition

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

A = QR

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

[Function] int gsl_linalg_QR_decomp (gsl matrix * A, gsl vector * tau)

This function factorizes the M-by-N matrix A into the QR decomposition A = QR. On output the diagonal and upper triangular part of the input matrix contain the matrix R. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and Householder vectors which encode the orthogonal matrix Q. The vector tau must be of length k = min(M, N). The matrix Q is related to these components by, Q = Q_{k}...Q_{2}Q_{1} where Q_{i} = I − τ_{i}v_{i}v_{i}^{T} and v_{i} is the Householder vector v_{i} = (0, ..., 1, A(i + 1, i), A(i + 2, i), ..., A(m, i)). This is the same storage scheme as used by lapack. The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan, Matrix Computations, Algorithm 5.2.1).

[Function] int gsl_linalg_QR_solve (const gsl matrix * QR, const gsl vector *

tau, const gsl vector * b, gsl vector * x) This function solves the square system Ax = b using the QR decomposition of A into (QR, tau) given by gsl_linalg_QR_decomp. The least-squares solution for rectangular systems can be found using gsl_linalg_QR_lssolve.

[Function] int gsl_linalg_QR_svx (const gsl matrix * QR, const gsl vector * tau,

gsl vector * x) This function solves the square system Ax = b in-place using the QR decomposition of A into (QR,tau) given by gsl_linalg_QR_decomp. On input x should contain the right-hand side b, which is replaced by the solution on output.

[Function] int gsl_linalg_QR_lssolve (const gsl matrix * QR, const gsl vector *

tau, const gsl vector * b, gsl vector * x, gsl vector * residual) This function finds the least squares solution to the overdetermined system Ax = b where the matrix A has more rows than columns. The least squares solution minimizes the Euclidean norm of the residual, ||Ax−b||.The routine uses the QR decomposition of A into (QR, tau) given by gsl_linalg_QR_decomp. The solution is returned in x. The residual is computed as a by-product and stored in residual.

[Function] int gsl_linalg_QR_QTvec (const gsl matrix * QR, const gsl vector *

tau, gsl vector * v) This function applies the matrix Q^{T} encoded in the decomposition (QR,tau) to the vector v, storing the result Q^{T} v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^{T}.

[Function] int gsl_linalg_QR_Qvec (const gsl matrix * QR, const gsl vector *

tau, gsl vector * v) This function applies the matrix Q encoded in the decomposition (QR,tau) to the vector v, storing the result Qv in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q.

[Function] int gsl_linalg_QR_Rsolve (const gsl matrix * QR, const gsl vector *

b, gsl vector * x) This function solves the triangular system Rx = b for x. It may be useful if the product b′ = Q^{T} b has already been computed using gsl_linalg_QR_QTvec.

[Function] int gsl_linalg_QR_Rsvx (const gsl matrix * QR, gsl vector * x)

This function solves the triangular system Rx = b for x in-place. On input x should contain the right-hand side b and is replaced by the solution on output. This function may be useful if the product b′ = Q^{T} b has already been computed using gsl_linalg_

QR_QTvec.

[Function] int gsl_linalg_QR_unpack (const gsl matrix * QR, const gsl vector *

tau, gsl matrix * Q, gsl matrix * R) This function unpacks the encoded QR decomposition (QR,tau) into the matrices Q and R, where Q is M-by-M and R is M-by-N.

[Function] int gsl_linalg_QR_QRsolve (gsl matrix * Q, gsl matrix * R, const

gsl vector * b, gsl vector * x) This function solves the system Rx = Q^{T} b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q, R).

[Function] int gsl_linalg_QR_update (gsl matrix * Q, gsl matrix * R, gsl vector *

w, const gsl vector * v) This function performs a rank-1 update wv^{T} of the QR decomposition (Q, R). The update is given by Q′R′ = QR + wv^{T} where the output matrices Q′ and R′ are also orthogonal and right triangular. Note that w is destroyed by the update.

[Function] int gsl_linalg_R_solve (const gsl matrix * R, const gsl vector * b,

gsl vector * x) This function solves the triangular system Rx = b for the N-by-N matrix R.

[Function] int gsl_linalg_R_svx (const gsl matrix * R, gsl vector * x)

This function solves the triangular system Rx = b in-place. On input x should contain the right-hand side b, which is replaced by the solution on output.

13.3 QR Decomposition with Column Pivoting

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

AP = QR

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

[Function] int gsl_linalg_QRPT_decomp (gsl matrix * A, gsl vector * tau,

gsl permutation * p, int * signum, gsl vector * norm) This function factorizes the M-by-N matrix A into the QRP^{T} decomposition A = QRP^{T}. On output the diagonal and upper triangular part of the input matrix contain the matrix R. The permutation matrix P is stored in the permutation p. The sign of the permutation is given by signum. It has the value (−1)^{n}, where n is the number of interchanges in the permutation. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and vectors which encode the orthogonal matrix Q. The vector tau must be of length k = min(M, N). The matrix Q is related to these components by, Q = Q_{k}...Q_{2}Q_{1} where Q_{i} = I−τ_{i}v_{i}v_{i}^{T} and v_{i} is the Householder vector v_{i} = (0, ..., 1, A(i + 1, i), A(i + 2, i), ..., A(m, i)). This is the same storage scheme as used by lapack. The vector norm is a workspace of length N used for column pivoting. The algorithm used to perform the decomposition is Householder QR with column pivoting (Golub & Van Loan, Matrix Computations, Algorithm 5.4.1).

[Function] int gsl_linalg_QRPT_decomp2 (const gsl matrix * A, gsl matrix * q,

gsl matrix * r, gsl vector * tau, gsl permutation * p, int * signum, gsl vector

   * norm)

This function factorizes the matrix A into the decomposition A = QRP^{T} without modifying A itself and storing the output in the separate matrices q and r.

[Function] int gsl_linalg_QRPT_solve (const gsl matrix * QR, const gsl vector *

tau, const gsl permutation * p, const gsl vector * b, gsl vector * x) This function solves the square system Ax = b using the QRP^{T} decomposition of A

into (QR, tau, p) given by gsl_linalg_QRPT_decomp.

[Function] int gsl_linalg_QRPT_svx (const gsl matrix * QR, const gsl vector *

tau, const gsl permutation * p, gsl vector * x) This function solves the square system Ax = b in-place using the QRP^{T} decomposition of A into (QR,tau,p). On input x should contain the right-hand side b, which is replaced by the solution on output.

[Function] int gsl_linalg_QRPT_QRsolve (const gsl matrix * Q, const gsl matrix

* R, const gsl permutation * p, const gsl vector * b, gsl vector * x) This function solves the square system RP^{T} x = Q^{T} b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q, R).

[Function] int gsl_linalg_QRPT_update (gsl matrix * Q, gsl matrix * R, const

gsl permutation * p, gsl vector * u, const gsl vector * v) This function performs a rank-1 update wv^{T} of the QRP^{T} decomposition (Q, R, p). The update is given by Q′R′ = QR + wv^{T} where the output matrices Q′ and R′ are also orthogonal and right triangular. Note that w is destroyed by the update. The permutation p is not changed.

[Function] int gsl_linalg_QRPT_Rsolve (const gsl matrix * QR, const

gsl permutation * p, const gsl vector * b, gsl vector * x) This function solves the triangular system RP^{T} x = b for the N-by-N matrix R contained in QR.

[Function] int gsl_linalg_QRPT_Rsvx (const gsl matrix * QR, const

gsl permutation * p, gsl vector * x) This function solves the triangular system RP^{T} x = b in-place for the N-by-N matrix R contained in QR. On input x should contain the right-hand side b, which is replaced by the solution on output.

13.4 Singular Value Decomposition

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

A = USV^{T}

The singular values σ_{i} = S_{ii} are all non-negative and are generally chosen to form a nonincreasing sequence σ_{1} ≥ σ_{2} ≥ ... ≥ σ_{N} ≥ 0.

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

[Function] int gsl_linalg_SV_decomp (gsl matrix * A, gsl matrix * V, gsl vector *

S, gsl vector * work) This function factorizes the M-by-N matrix A into the singular value decomposition A = USV^{T} for M ≥ N. On output the matrix A is replaced by U. The diagonal elements of the singular value matrix S are stored in the vector S. The singular values are non-negative and form a non-increasing sequence from S_{1} to S_{N}. The matrix V contains the elements of V in untransposed form. To form the product USV^{T} it is necessary to take the transpose of V. A workspace of length N is required in work. This routine uses the Golub-Reinsch SVD algorithm.

[Function] int gsl_linalg_SV_decomp_mod (gsl matrix * A, gsl matrix * X,

gsl matrix * V, gsl vector * S, gsl vector * work) This function computes the SVD using the modified Golub-Reinsch algorithm, which is faster for M ≫ N. It requires the vector work of length N and the N-by-N matrix X as additional working space.

[Function] int gsl_linalg_SV_decomp_jacobi (gsl matrix * A, gsl matrix * V,

gsl vector * S) This function computes the SVD of the M-by-N matrix A using one-sided Jacobi orthogonalization for M ≥ N. The Jacobi method can compute singular values to higher relative accuracy than Golub-Reinsch algorithms (see references for details).

[Function] int gsl_linalg_SV_solve (gsl matrix * U, gsl matrix * V, gsl vector *

S, const gsl vector * b, gsl vector * x) This function solves the system Ax = b using the singular value decomposition (U, S,

V) of A given by gsl_linalg_SV_decomp.

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

In the over-determined case where A has more rows than columns the system is solved in the least squares sense, returning the solution x which minimizes ||Ax − b||_{2}.

13.5 Cholesky Decomposition

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

A = LL^{T}

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

[Function] int gsl_linalg_cholesky_decomp (gsl matrix * A)

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

code GSL_EDOM.
[Function] int gsl_linalg_cholesky_solve (const gsl matrix * cholesky, const

gsl vector * b, gsl vector * x) This function solves the system Ax = b using the Cholesky decomposition of A into

the matrix cholesky given by gsl_linalg_cholesky_decomp.
[Function] int gsl_linalg_cholesky_svx (const gsl matrix * cholesky,

gsl vector * x) This function solves the system Ax = b in-place using the Cholesky decomposition of A into the matrix cholesky given by gsl_linalg_cholesky_decomp. On input x should contain the right-hand side b, which is replaced by the solution on output.

13.6 Tridiagonal Decomposition of Real Symmetric Matrices

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

A = QT Q^{T}

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

[Function] int gsl_linalg_symmtd_decomp (gsl matrix * A, gsl vector * tau)

This function factorizes the symmetric square matrix A into the symmetric tridiagonal decomposition QT Q^{T}. On output the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the orthogonal matrix Q. This storage scheme is the same as used by lapack. The upper triangular part of A is not referenced.

[Function] int gsl_linalg_symmtd_unpack (const gsl matrix * A, const gsl vector

* tau, gsl matrix * Q, gsl vector * diag, gsl vector * subdiag) This function unpacks the encoded symmetric tridiagonal decomposition (A, tau) obtained from gsl_linalg_symmtd_decomp into the orthogonal matrix Q, the vector of diagonal elements diag and the vector of subdiagonal elements subdiag.

[Function] int gsl_linalg_symmtd_unpack_T (const gsl matrix * A, gsl vector *

diag, gsl vector * subdiag) This function unpacks the diagonal and subdiagonal of the encoded symmetric tridiagonal decomposition (A, tau) obtained from gsl_linalg_symmtd_decomp into the vectors diag and subdiag.

13.7 Tridiagonal Decomposition of Hermitian Matrices

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

A = UT U^{T}

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

[Function] int gsl_linalg_hermtd_decomp (gsl matrix complex * A,

gsl vector complex * tau) This function factorizes the hermitian matrix A into the symmetric tridiagonal decomposition UT U^{T}. On output the real parts of the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the orthogonal matrix Q. This storage scheme is the same as used by lapack. The upper triangular part of A and imaginary parts of the diagonal are not referenced.

[Function] int gsl_linalg_hermtd_unpack (const gsl matrix complex * A, const gsl vector complex * tau, gsl matrix complex * Q, gsl vector * diag, gsl vector * subdiag)

This function unpacks the encoded tridiagonal decomposition (A, tau) obtained from gsl_linalg_hermtd_decomp into the unitary matrix U, the real vector of diagonal elements diag and the real vector of subdiagonal elements subdiag.

[Function] int gsl_linalg_hermtd_unpack_T (const gsl matrix complex * A,

gsl vector * diag, gsl vector * subdiag) This function unpacks the diagonal and subdiagonal of the encoded tridiagonal decomposition (A, tau) obtained from the gsl_linalg_hermtd_decomp into the real vectors diag and subdiag.

13.8 Bidiagonalization

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

A = UBV^{T}

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

[Function] int gsl_linalg_bidiag_decomp (gsl matrix * A, gsl vector * tau_U,

gsl vector * tau_V) This function factorizes the M-by-N matrix A into bidiagonal form UBV^{T}. The diagonal and superdiagonal of the matrix B are stored in the diagonal and superdiagonal of A. The orthogonal matrices U and V are stored as compressed Householder vectors in the remaining elements of A. The Householder coefficients are stored in the vectors tau U and tau V. The length of tau U must equal the number of elements in the diagonal of A and the length of tau V should be one element shorter.

[Function] int gsl_linalg_bidiag_unpack (const gsl matrix * A, const gsl vector * tau_U, gsl matrix * U, const gsl vector * tau_V, gsl matrix * V, gsl vector *

   diag, gsl vector * superdiag)

This function unpacks the bidiagonal decomposition of A given by gsl_linalg_ bidiag_decomp, (A, tau U, tau V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag. Note that U is stored as a compact M-by-N orthogonal matrix satisfying U^{T} U = I for efficiency.

[Function] int gsl_linalg_bidiag_unpack2 (gsl matrix * A, gsl vector * tau_U,

gsl vector * tau_V, gsl matrix * V) This function unpacks the bidiagonal decomposition of A given by gsl_linalg_ bidiag_decomp, (A, tau U, tau V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag. The matrix U is stored in-place in A.

[Function] int gsl_linalg_bidiag_unpack_B (const gsl matrix * A, gsl vector *

   diag, gsl vector * superdiag)

This function unpacks the diagonal and superdiagonal of the bidiagonal decomposition of A given by gsl_linalg_bidiag_decomp, into the diagonal vector diag and superdiagonal vector superdiag.

13.9 Householder Transformations

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

P = I − τvv^{T}

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

[Function] double gsl_linalg_householder_transform (gsl vector * v)

This function prepares a Householder transformation P = I − τvv^{T} which can be used to zero all the elements of the input vector except the first. On output the transformation is stored in the vector v and the scalar τ is returned.

[Function] int gsl_linalg_householder_hm (double tau, const gsl vector * v,

gsl matrix * A) This function applies the Householder matrix P defined by the scalar tau and the vector v to the left-hand side of the matrix A. On output the result P A is stored in A.

[Function] int gsl_linalg_householder_mh (double tau, const gsl vector * v,

gsl matrix * A) This function applies the Householder matrix P defined by the scalar tau and the vector v to the right-hand side of the matrix A. On output the result AP is stored in A.

[Function] int gsl_linalg_householder_hv (double tau, const gsl vector * v,

gsl vector * w) This function applies the Householder transformation P defined by the scalar tau and the vector v to the vector w. On output the result P w is stored in w.

13.10 Householder solver for linear systems

[Function] int gsl_linalg_HH_solve (gsl matrix * A, const gsl vector * b,

gsl vector * x) This function solves the system Ax = b directly using Householder transformations. On output the solution is stored in x and b is not modified. The matrix A is destroyed by the Householder transformations.

[Function] int gsl_linalg_HH_svx (gsl matrix * A, gsl vector * x)

This function solves the system Ax = b in-place using Householder transformations. On input x should contain the right-hand side b, which is replaced by the solution on output. The matrix A is destroyed by the Householder transformations.

13.11 Tridiagonal Systems

[Function] int gsl_linalg_solve_tridiag (const gsl vector * diag, const

gsl vector * e, const gsl vector * f, const gsl vector * b, gsl vector * x) This function solves the general N-by-N system Ax = b where A is tridiagonal (N ≥ 2). The super-diagonal and sub-diagonal vectors e and f must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( \begin{matrix}d_{0} & e_{0} & 0 & 0 \\ f_{0} & d_{1} & e_{1} & 0 \\ 0 & f_{1} & d_{2} & e_{2} \\ 0 & 0 & f_{2} & d_{3} \end{matrix} )

[Function] int gsl_linalg_solve_symm_tridiag (const gsl vector * diag, const

gsl vector * e, const gsl vector * b, gsl vector * x) This function solves the general N-by-N system Ax = b where A is symmetric tridiagonal (N ≥ 2). The off-diagonal vector e must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( \begin{matrix}d_{0} & e_{0} & 0 & 0 \\ e_{0} & d_{1} & e_{1} & 0 \\ 0 & e_{1} & d_{2} & e_{2} \\ 0 & 0 & e_{2} & d_{3} \end{matrix} )

[Function] int gsl_linalg_solve_cyc_tridiag (const gsl vector * diag, const

gsl vector * e, const gsl vector * f, const gsl vector * b, gsl vector * x) This function solves the general N-by-N system Ax = b where A is cyclic tridiagonal (N ≥ 3). The cyclic super-diagonal and sub-diagonal vectors e and f must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( \begin{matrix}d_{0} & e_{0} & 0 & f_{3} \\ f_{0} & d_{1} & e_{1} & 0 \\ 0 & f_{1} & d_{2} & e_{2} \\ e_{3} & 0 & f_{2} & d_{3} \end{matrix} )

[Function] int gsl_linalg_solve_symm_cyc_tridiag (const gsl vector * diag,

const gsl vector * e, const gsl vector * b, gsl vector * x) This function solves the general N-by-N system Ax = b where A is symmetric cyclic tridiagonal (N ≥ 3). The cyclic off-diagonal vector e must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( \begin{matrix}d_{0} & e_{0} & 0 & e_{3} \\ e_{0} & d_{1} & e_{1} & 0 \\ 0 & e_{1} & d_{2} & e_{2} \\ e_{3} & 0 & e_{2} & d_{3} \end{matrix} )

13.12 Examples

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

( \begin{matrix}0.18 & 0.60 & 0.57 & 0.96 \\ 0.41 & 0.24 & 0.99 & 0.58 \\ 0.14 & 0.30 & 0.97 & 0.66 \\ 0.51 & 0.13 & 0.19 & 0.85 \end{matrix} ) ( \begin{matrix}x_{0} \\ x_{1} \\ x_{2} \\ x_{3} \end{matrix} ) = ( \begin{matrix}1.0 \\ 2.0 \\ 3.0 \\ 4.0 \end{matrix} )

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

#include <stdio.h>
#include <gsl/gsl_linalg.h>
int
main (void)
{

double a_data[] = { 0.18, 0.60, 0.57, 0.96,
                    0.41, 0.24, 0.99, 0.58,
                    0.14, 0.30, 0.97, 0.66,
                    0.51, 0.13, 0.19, 0.85 };
double b_data[] = { 1.0, 2.0, 3.0, 4.0 };
gsl_matrix_view m
  = gsl_matrix_view_array (a_data, 4, 4);
gsl_vector_view b
  = gsl_vector_view_array (b_data, 4);
gsl_vector *x = gsl_vector_alloc (4);
int s;
gsl_permutation * p = gsl_permutation_alloc (4);
gsl_linalg_LU_decomp (&m.matrix, p, &s);
gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);
printf ("x = \n");
gsl_vector_fprintf (stdout, x, "%g");
gsl_permutation_free (p);
return 0;
}

Here is the output from the program,

x = -4.05205
-12.6056
1.66091
8.69377

This can be verified by multiplying the solution x by the original matrix A using gnu octave,

octave> A = [ 0.18, 0.60, 0.57, 0.96;
            0.41, 0.24, 0.99, 0.58;
            0.14, 0.30, 0.97, 0.66;
            0.51, 0.13, 0.19, 0.85 ];
octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];
octave> A * x
ans =
1.0000

   2.0000
   3.0000
   4.0000

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

13.13 References and Further Reading

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

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

The lapack library is described in the following manual, LAPACK Users’ Guide (Third Edition, 1999), Published by SIAM, ISBN 0-89871-447-8.

http://www.netlib.org/lapack

The lapack source code can be found at the website above, along with an online copy of the users guide. The Modified Golub-Reinsch algorithm is described in the following paper,

T.F. Chan, “An Improved Algorithm for Computing the Singular Value Decomposition”, ACM Transactions on Mathematical Software, 8 (1982), pp 72–83.

The Jacobi algorithm for singular value decomposition is described in the following papers, J.C. Nash, “A one-sided transformation method for the singular value decomposition and algebraic eigenproblem”, Computer Journal, Volume 18, Number 1 (1973), p 74–76 James Demmel, Kresimir Veselic, “Jacobi’s Method is more accurate than QR”, Lapack Working Note 15 (LAWN-15), October 1989. Available from netlib,

http://www.netlib.org/lapack/ in the lawns or lawnspdf directories.

14 Eigensystems

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

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

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

14.1 Real Symmetric Matrices

[Function] gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const size t
   n)

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

[Function] void gsl_eigen_symm_free (gsl eigen symm workspace * w)

This function frees the memory associated with the workspace w.

[Function] int gsl_eigen_symm (gsl matrix * A, gsl vector * eval,

gsl eigen symm workspace * w) This function computes the eigenvalues of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered.

[Function] gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const

size t n) This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real symmetric matrices. The size of the workspace is O(4n).

[Function] void gsl_eigen_symmv_free (gsl eigen symmv workspace * w)

This function frees the memory associated with the workspace w.

[Function] int gsl_eigen_symmv (gsl matrix * A, gsl vector * eval, gsl matrix *

evec, gsl eigen symmv workspace * w) This function computes the eigenvalues and eigenvectors of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding eigenvectors are stored in the columns of the matrix evec. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.

14.2 Complex Hermitian Matrices

[Function] gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const size t
   n)

This function allocates a workspace for computing eigenvalues of n-by-n complex hermitian matrices. The size of the workspace is O(3n).

[Function] void gsl_eigen_herm_free (gsl eigen herm workspace * w)

This function frees the memory associated with the workspace w.

[Function] int gsl_eigen_herm (gsl matrix complex * A, gsl vector * eval,

gsl eigen herm workspace * w) This function computes the eigenvalues of the complex hermitian matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered.

[Function] gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const

size t n) This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex hermitian matrices. The size of the workspace is O(5n).

[Function] void gsl_eigen_hermv_free (gsl eigen hermv workspace * w)

This function frees the memory associated with the workspace w.

[Function] int gsl_eigen_hermv (gsl matrix complex * A, gsl vector * eval,

gsl matrix complex * evec, gsl eigen hermv workspace * w) This function computes the eigenvalues and eigenvectors of the complex hermitian matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding complex eigenvectors are stored in the columns of the matrix evec. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.

14.3 Sorting Eigenvalues and Eigenvectors

[Function] int gsl_eigen_symmv_sort (gsl vector * eval, gsl matrix * evec,

gsl eigen sort t sort_type) This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort type,

GSL_EIGEN_SORT_VAL_ASC

ascending order in numerical value

GSL_EIGEN_SORT_VAL_DESC

descending order in numerical value

GSL_EIGEN_SORT_ABS_ASC

ascending order in magnitude

GSL_EIGEN_SORT_ABS_DESC

descending order in magnitude

[Function] int gsl_eigen_hermv_sort (gsl vector * eval, gsl matrix complex *

evec, gsl eigen sort t sort_type) This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding complex eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort type as shown above.

14.4 Examples

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

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>
int
main (void)
{
  double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
                    1/2.0, 1/3.0, 1/4.0, 1/5.0,
                    1/3.0, 1/4.0, 1/5.0, 1/6.0,
                    1/4.0, 1/5.0, 1/6.0, 1/7.0 };
  gsl_matrix_view m
    = gsl_matrix_view_array (data, 4, 4);
  gsl_vector *eval = gsl_vector_alloc (4);
  gsl_matrix *evec = gsl_matrix_alloc (4, 4);
  gsl_eigen_symmv_workspace * w =
    gsl_eigen_symmv_alloc (4);
  gsl_eigen_symmv (&m.matrix, eval, evec, w);
  gsl_eigen_symmv_free (w);
  gsl_eigen_symmv_sort (eval, evec,
                        GSL_EIGEN_SORT_ABS_ASC);
  {

    int i;
    for (i = 0; i < 4; i++)
      {
        double eval_i
           = gsl_vector_get (eval, i);
        gsl_vector_view evec_i
           = gsl_matrix_column (evec, i);
        printf ("eigenvalue = %g\n", eval_i);
        printf ("eigenvector = \n");
        gsl_vector_fprintf (stdout,
                            &evec_i.vector, "%g");
      }
  }
  return 0;
}

Here is the beginning of the output from the program,

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

This can be compared with the corresponding output from gnu octave,

octave> [v,d] = eig(hilb(4));
octave> diag(d)
ans =
   9.6702e-05
   6.7383e-03
   1.6914e-01
   1.5002e+00
octave> v
v =
   0.029193   0.179186  -0.582076   0.792608
  -0.328712  -0.741918   0.370502   0.451923
   0.791411   0.100228   0.509579   0.322416
  -0.514553   0.638283   0.514048   0.252161

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

14.5 References and Further Reading

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

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

The lapack library is described in, LAPACK Users’ Guide (Third Edition, 1999), Published by SIAM, ISBN 0-89871-447-8.

    http://www.netlib.org/lapack

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

15 Fast Fourier Transforms (FFTs)

This chapter describes functions for performing Fast Fourier Transforms (FFTs). The library includes radix-2 routines (for lengths which are a power of two) and mixed-radix routines (which work for any length). For efficiency there are separate versions of the routines for real data and for complex data. The mixed-radix routines are a reimplementation of the fftpack library of Paul Swarztrauber. Fortran code for fftpack is available on Netlib (fftpack also includes some routines for sine and cosine transforms but these are currently not available in GSL). For details and derivations of the underlying algorithms consult the document GSL FFT Algorithms (see Section 15.8 [FFT References and Further Reading], page 155)

15.1 Mathematical Definitions

Fast Fourier Transforms are efficient algorithms for calculating the discrete fourier transform (DFT),

x_{j} = ∑_{k=0}^{N−1} z_{k} exp(−2πijk/N)

The DFT usually arises as an approximation to the continuous fourier transform when functions are sampled at discrete intervals in space or time. The naive evaluation of the discrete fourier transform is a matrix-vector multiplication W z⃗. A general matrix-vector multiplication takes O(N^{2}) operations for N data-points. Fast fourier transform algorithms use a divide-and-conquer strategy to factorize the matrix W into smaller sub-matrices, corresponding to the integer factors of the length N. If N can be factorized into a product of integers f_{1}f_{2} … f_{n} then the DFT can be computed in O(N ∑ f_{i}) operations. For a radix-2 FFT this gives an operation count of O(N log_{2} N).

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

x_{j} = ∑_{k=0}^{N−1} z_{k} exp(−2πijk/N)

and the definition of the inverse fourier transform, x = IFFT(z), is,

z_{j} = \frac{1}{N} ∑_{k=0}^{N−1} x_{k} exp(2πijk/N).

The factor of 1/N makes this a true inverse. For example, a call to gsl_fft_complex_ forward followed by a call to gsl_fft_complex_inverse should return the original data (within numerical errors).

In general there are two possible choices for the sign of the exponential in the transform/ inverse-transform pair. GSL follows the same convention as fftpack, using a negative exponential for the forward transform. The advantage of this convention is that the inverse transform recreates the original function with simple fourier synthesis. Numerical Recipes uses the opposite convention, a positive exponential in the forward transform.

The backwards FFT is simply our terminology for an unscaled version of the inverse FFT,

z_{j}^{backwards} = ∑_{k=0}^{N−1} x_{k} exp(2πijk/N).

When the overall scale of the result is unimportant it is often convenient to use the backwards FFT instead of the inverse to save unnecessary divisions.

15.2 Overview of complex data FFTs

The inputs and outputs for the complex FFT routines are packed arrays of floating point numbers. In a packed array the real and imaginary parts of each complex number are placed in alternate neighboring elements. For example, the following definition of a packed array of length 6,

     double x[3*2];
     gsl_complex_packed_array data = x;

can be used to hold an array of three complex numbers, z[3], in the following way,

     data[0] = Re(z[0])
     data[1] = Im(z[0])
     data[2] = Re(z[1])
     data[3] = Im(z[1])
     data[4] = Re(z[2])
     data[5] = Im(z[2])

The array indices for the data have the same ordering as those in the definition of the DFT—i.e. there are no index transformations or permutations of the data.

A stride parameter allows the user to perform transforms on the elements z[stride*i] instead of z[i]. A stride greater than 1 can be used to take an in-place FFT of the column of a matrix. A stride of 1 accesses the array without any additional spacing between elements.

To perform an FFT on a vector argument, such as gsl_complex_vector * v, use the following definitions (or their equivalents) when calling the functions described in this chapter:

     gsl_complex_packed_array data = v->data;
     size_t stride = v->stride;
     size_t n = v->size;

For physical applications it is important to remember that the index appearing in the DFT does not correspond directly to a physical frequency. If the time-step of the DFT is Δ then the frequency-domain includes both positive and negative frequencies, ranging from −1/(2Δ) through 0 to +1/(2Δ). The positive frequencies are stored from the beginning of the array up to the middle, and the negative frequencies are stored backwards from the end of the array.

Here is a table which shows the layout of the array data, and the correspondence between the time-domain data z, and the frequency-domain data x.

     index    z               x = FFT(z)
     0        z(t = 0)        x(f = 0)

1        z(t = 1)        x(f = 1/(N Delta))
2        z(t = 2)        x(f = 2/(N Delta))
.        ........        ..................
N/2      z(t = N/2)      x(f = +1/(2 Delta),
                               -1/(2 Delta))
.        ........        ..................
N-3      z(t = N-3)      x(f = -3/(N Delta))
N-2      z(t = N-2)      x(f = -2/(N Delta))
N-1      z(t = N-1)      x(f = -1/(N Delta))

When N is even the location N/2 contains the most positive and negative frequencies (+1/(2Δ), −1/(2Δ)) which are equivalent. If N is odd then general structure of the table above still applies, but N/2 does not appear.

15.3 Radix-2 FFT routines for complex data

The radix-2 algorithms described in this section are simple and compact, although not necessarily the most efficient. They use the Cooley-Tukey algorithm to compute in-place complex FFTs for lengths which are a power of 2—no additional storage is required. The corresponding self-sorting mixed-radix routines offer better performance at the expense of requiring additional working space.

All the functions described in this section are declared in the header file

‘gsl_fft_complex.h’.
[Function] int gsl_fft_complex_radix2_forward (gsl complex packed array

data, size t stride, size t n)

[Function] int gsl_fft_complex_radix2_transform (gsl complex packed array

data, size t stride, size t n, gsl fft direction sign)

[Function] int gsl_fft_complex_radix2_backward (gsl complex packed array

data, size t stride, size t n)

[Function] int gsl_fft_complex_radix2_inverse (gsl complex packed array

data, size t stride, size t n) These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array data using an in-place radix-2 decimation-in-time algorithm. The length of the transform n is restricted to powers of two. For the transform version of the function the sign argument can be either forward (−1) or

backward (+1).

The functions return a value of GSL_SUCCESS if no errors were detected, or GSL_EDOM if the length of the data n is not a power of two.

[Function] int gsl_fft_complex_radix2_dif_forward

(gsl complex packed array data, size t stride, size t n)

[Function] int gsl_fft_complex_radix2_dif_transform

(gsl complex packed array data, size t stride, size t n, gsl fft direction

   sign)
[Function] int gsl_fft_complex_radix2_dif_backward

(gsl complex packed array data, size t stride, size t n)

[Function] int gsl_fft_complex_radix2_dif_inverse

(gsl complex packed array data, size t stride, size t n) These are decimation-in-frequency versions of the radix-2 FFT functions.

Here is an example program which computes the FFT of a short pulse in a sample of length 128. To make the resulting fourier transform real the pulse is defined for equal positive and negative times (−10 … 10), where the negative times wrap around the end of the array.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
  int i; double data[2*128];
  for (i = 0; i < 128; i++)
    {
       REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
    }
  REAL(data,0) = 1.0;
  for (i = 1; i <= 10; i++)
    {
       REAL(data,i) = REAL(data,128-i) = 1.0;
    }
  for (i = 0; i < 128; i++)
    {
      printf ("%d %e %e\n", i,
              REAL(data,i), IMAG(data,i));
    }
  printf ("\n");
  gsl_fft_complex_radix2_forward (data, 1, 128);
  for (i = 0; i < 128; i++)
    {
      printf ("%d %e %e\n", i,
              REAL(data,i)/sqrt(128),
              IMAG(data,i)/sqrt(128));

         }
       return 0;
     }

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

the return status of gsl_fft_complex_radix2_forward.

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

∫_{−a}^{+a} e^{−2πikx}dx = \frac{sin(2πka)}{πk}

A pulse and its discrete fourier transform, output from the example program.

15.4 Mixed-radix FFT routines for complex data

This section describes mixed-radix FFT algorithms for complex data. The mixed-radix functions work for FFTs of any length. They are a reimplementation of Paul Swarztrauber’s Fortran fftpack library. The theory is explained in the review article Self-sorting Mixedradix FFTs by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as fftpack.

The mixed-radix algorithm is based on sub-transform modules—highly optimized small length FFTs which are combined to create larger FFTs. There are efficient modules for factors of 2, 3, 4, 5, 6 and 7. The modules for the composite factors of 4 and 6 are faster than combining the modules for 2 ∗ 2 and 2 ∗ 3.

For factors which are not implemented as modules there is a fall-back to a general lengthn module which uses Singleton’s method for efficiently computing a DFT. This module is O(n^{2}), and slower than a dedicated module would be but works for any length n. Of course, lengths which use the general length-n module will still be factorized as much as possible. For example, a length of 143 will be factorized into 11∗13. Large prime factors are the worst case scenario, e.g. as found in n = 2 ∗ 3 ∗ 99991, and should be avoided because their O(n^{2}) scaling will dominate the run-time (consult the document GSL FFT Algorithms included in the GSL distribution if you encounter this problem).

The mixed-radix initialization function gsl_fft_complex_wavetable_alloc returns the list of factors chosen by the library for a given length N. It can be used to check how well the length has been factorized, and estimate the run-time. To a first approximation the run-time scales as N ∑ f_{i}, where the f_{i} are the factors of N. For programs under user control you may wish to issue a warning that the transform will be slow when the length is poorly factorized. If you frequently encounter data lengths which cannot be factorized using the existing small-prime modules consult GSL FFT Algorithms for details on adding support for other factors.

All the functions described in this section are declared in the header file

‘gsl_fft_complex.h’.
[Function] gsl_fft_complex_wavetable *
        gsl_fft_complex_wavetable_alloc (size t n)

This function prepares a trigonometric lookup table for a complex FFT of length n. The function returns a pointer to the newly allocated gsl_fft_complex_wavetable if no errors were detected, and a null pointer in the case of error. The length n is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to sin and cos, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then this computation is a one-off overhead which does not affect the final throughput. The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (or inverse) transforms of a given length.

[Function] void gsl_fft_complex_wavetable_free (gsl fft complex wavetable *
        wavetable)

This function frees the memory associated with the wavetable wavetable. The wavetable can be freed if no further FFTs of the same length will be needed.

These functions operate on a gsl_fft_complex_wavetable structure which contains internal parameters for the FFT. It is not necessary to set any of the components directly but it can sometimes be useful to examine them. For example, the chosen factorization of the FFT length is given and can be used to provide an estimate of the run-time or numerical error. The wavetable structure is declared in the header file ‘gsl_fft_complex.h’.

[Data Type] gsl_fft_complex_wavetable

This is a structure that holds the factorization and trigonometric lookup tables for the mixed radix fft algorithm. It has the following components:

size_t n This is the number of complex data points

size_t nf This is the number of factors that the length n was decomposed into.

size_t factor[64]

This is the array of factors. Only the first nf elements are used.

gsl_complex * trig

This is a pointer to a preallocated trigonometric lookup table of n complex elements.

gsl_complex * twiddle[64]

This is an array of pointers into trig, giving the twiddle factors for each pass.

The mixed radix algorithms require additional working space to hold the intermediate steps of the transform.

[Function] gsl_fft_complex_workspace *
   gsl_fft_complex_workspace_alloc (size t n)

This function allocates a workspace for a complex transform of length n.

[Function] void gsl_fft_complex_workspace_free (gsl fft complex workspace *
   workspace)

This function frees the memory associated with the workspace workspace. The workspace can be freed if no further FFTs of the same length will be needed.

The following functions compute the transform,

[Function] int gsl_fft_complex_forward (gsl complex packed array data, size t

stride, size t n, const gsl fft complex wavetable * wavetable, gsl fft complex workspace * work)

[Function] int gsl_fft_complex_transform (gsl complex packed array data,

size t stride, size t n, const gsl fft complex wavetable * wavetable, gsl fft complex workspace * work, gsl fft direction sign)

[Function] int gsl_fft_complex_backward (gsl complex packed array data,

size t stride, size t n, const gsl fft complex wavetable * wavetable, gsl fft complex workspace * work)

[Function] int gsl_fft_complex_inverse (gsl complex packed array data, size t

stride, size t n, const gsl fft complex wavetable * wavetable, gsl fft complex workspace * work)

These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array data, using a mixed radix decimation-in-frequency algorithm. There is no restriction on the length n. Efficient modules are provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining factors are computed with a slow, O(n^{2}), general-n module. The caller must supply a wavetable containing the trigonometric lookup tables and a workspace work. For the transform version of the function the sign argument can be either forward (−1) or backward (+1). The functions return a value of 0 if no errors were detected. The following gsl_errno conditions are defined for these functions:

GSL_EDOM The length of the data n is not a positive integer (i.e. n is zero).

GSL_EINVAL

The length of the data n and the length used to compute the given wavetable do not match.

Here is an example program which computes the FFT of a short pulse in a sample of length 630 (= 2 ∗ 3 ∗ 3 ∗ 5 ∗ 7) using the mixed-radix algorithm.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
  int i;
  const int n = 630;
  double data[2*n];
  gsl_fft_complex_wavetable * wavetable;
  gsl_fft_complex_workspace * workspace;
  for (i = 0; i < n; i++)
    {
      REAL(data,i) = 0.0;
      IMAG(data,i) = 0.0;
    }
  data[0] = 1.0;
  for (i = 1; i <= 10; i++)
    {
      REAL(data,i) = REAL(data,n-i) = 1.0;
    }
  for (i = 0; i < n; i++)
    {
      printf ("%d: %e %e\n", i, REAL(data,i),

                                     IMAG(data,i));
         }
       printf ("\n");
       wavetable = gsl_fft_complex_wavetable_alloc (n);
       workspace = gsl_fft_complex_workspace_alloc (n);
       for (i = 0; i < wavetable->nf; i++)
         {
            printf ("# factor %d: %d\n", i,
                    wavetable->factor[i]);
         }
       gsl_fft_complex_forward (data, 1, n,
                                wavetable, workspace);
       for (i = 0; i < n; i++)
         {
           printf ("%d: %e %e\n", i, REAL(data,i),
                                     IMAG(data,i));
         }
       gsl_fft_complex_wavetable_free (wavetable);
       gsl_fft_complex_workspace_free (workspace);
       return 0;
     }

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

15.5 Overview of real data FFTs

The functions for real data are similar to those for complex data. However, there is an important difference between forward and inverse transforms. The fourier transform of a real sequence is not real. It is a complex sequence with a special symmetry:

z_{k} = z_{N−k}^{∗}

A sequence with this symmetry is called conjugate-complex or half-complex. This different structure requires different storage layouts for the forward transform (from real to halfcomplex) and inverse transform (from half-complex back to real). As a consequence the routines are divided into two sets: functions in gsl_fft_real which operate on real sequences and functions in gsl_fft_halfcomplex which operate on half-complex sequences.

Functions in gsl_fft_real compute the frequency coefficients of a real sequence. The half-complex coefficients c of a real sequence x are given by fourier analysis,

c_{k} = ∑_{j=0}^{N−1} x_{j} exp(−2πijk/N)

Functions in gsl_fft_halfcomplex compute inverse or backwards transforms. They reconstruct real sequences by fourier synthesis from their half-complex frequency coefficients, c,

x_{j} = \frac{1}{N} ∑_{k=0}^{N−1} c_{k} exp(2πijk/N)

The symmetry of the half-complex sequence implies that only half of the complex numbers in the output need to be stored. The remaining half can be reconstructed using the halfcomplex symmetry condition. This works for all lengths, even and odd—when the length is even the middle value where k = N/2 is also real. Thus only N real numbers are required to store the half-complex sequence, and the transform of a real sequence can be stored in the same size array as the original data.

The precise storage arrangements depend on the algorithm, and are different for radix-2 and mixed-radix routines. The radix-2 function operates in-place, which constrains the locations where each element can be stored. The restriction forces real and imaginary parts to be stored far apart. The mixed-radix algorithm does not have this restriction, and it stores the real and imaginary parts of a given term in neighboring locations (which is desirable for better locality of memory accesses).

15.6 Radix-2 FFT routines for real data

This section describes radix-2 FFT algorithms for real data. They use the Cooley-Tukey algorithm to compute in-place FFTs for lengths which are a power of 2.

The radix-2 FFT functions for real data are declared in the header files ‘gsl_fft_real.h’

[Function] int gsl_fft_real_radix2_transform (double data[], size t stride,

size t n) This function computes an in-place radix-2 FFT of length n and stride stride on the real array data. The output is a half-complex sequence, which is stored in-place. The arrangement of the half-complex terms uses the following scheme: for k < N/2 the real part of the k-th term is stored in location k, and the corresponding imaginary part is stored in location N − k. Terms with k > N/2 can be reconstructed using the symmetry z_{k} = z_{N−k}^{∗}. The terms for k = 0 and k = N/2 are both purely real, and count as a special case. Their real parts are stored in locations 0 and N/2 respectively, while their imaginary parts which are zero are not stored. The following table shows the correspondence between the output data and the equivalent results obtained by considering the input data as a complex sequence with zero imaginary part,

          complex[0].real    =    data[0]
          complex[0].imag    =    0
          complex[1].real    =    data[1]
          complex[1].imag    =    data[N-1]
          ...............         ................
          complex[k].real    =    data[k]
          complex[k].imag    =    data[N-k]
          ...............         ................

          complex[N/2].real  =    data[N/2]
          complex[N/2].imag  =    0
          ...............         ................
          complex[k'].real   =    data[k]        k' = N - k
          complex[k'].imag   =   -data[N-k]
          ...............         ................
          complex[N-1].real  =    data[1]
          complex[N-1].imag  =   -data[N-1]

The radix-2 FFT functions for halfcomplex data are declared in the header file

‘gsl_fft_halfcomplex.h’.
[Function] int gsl_fft_halfcomplex_radix2_inverse (double data[], size t
        stride, size t n)
[Function] int gsl_fft_halfcomplex_radix2_backward (double data[], size t
        stride, size t n)

These functions compute the inverse or backwards in-place radix-2 FFT of length n and stride stride on the half-complex sequence data stored according the output scheme used by gsl_fft_real_radix2. The result is a real array stored in natural order.

15.7 Mixed-radix FFT routines for real data

This section describes mixed-radix FFT algorithms for real data. The mixed-radix functions work for FFTs of any length. They are a reimplementation of the real-FFT routines in the Fortran fftpack library by Paul Swarztrauber. The theory behind the algorithm is explained in the article Fast Mixed-Radix Real Fourier Transforms by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as fftpack.

The functions use the fftpack storage convention for half-complex sequences. In this convention the half-complex transform of a real sequence is stored with frequencies in increasing order, starting at zero, with the real and imaginary parts of each frequency in neighboring locations. When a value is known to be real the imaginary part is not stored. The imaginary part of the zero-frequency component is never stored. It is known to be zero (since the zero frequency component is simply the sum of the input data (all real)). For a sequence of even length the imaginary part of the frequency n/2 is not stored either, since the symmetry z_{k} = z_{N−k}^{∗} implies that this is purely real too.

The storage scheme is best shown by some examples. The table below shows the output for an odd-length sequence, n = 5. The two columns give the correspondence between the 5 values in the half-complex sequence returned by gsl_fft_real_transform, halfcomplex[] and the values complex[] that would be returned if the same real input sequence were passed to gsl_fft_complex_backward as a complex sequence (with imaginary parts set to 0),

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]

complex[3].real  =  halfcomplex[3]
complex[3].imag  = -halfcomplex[4]
complex[4].real  =  halfcomplex[1]
complex[4].imag  = -halfcomplex[2]

The upper elements of the complex array, complex[3] and complex[4] are filled in using the symmetry condition. The imaginary part of the zero-frequency term complex[0].imag is known to be zero by the symmetry.

The next table shows the output for an even-length sequence, n = 6 In the even case there are two values which are purely real,

complex[0].real  =  halfcomplex[0]
complex[0].imag  =  0
complex[1].real  =  halfcomplex[1]
complex[1].imag  =  halfcomplex[2]
complex[2].real  =  halfcomplex[3]
complex[2].imag  =  halfcomplex[4]
complex[3].real  =  halfcomplex[5]
complex[3].imag  =  0
complex[4].real  =  halfcomplex[3]
complex[4].imag  = -halfcomplex[4]
complex[5].real  =  halfcomplex[1]
complex[5].imag  = -halfcomplex[2]

The upper elements of the complex array, complex[4] and complex[5] are filled in using the symmetry condition. Both complex[0].imag and complex[3].imag are known to be zero.

All these functions are declared in the header files ‘gsl_fft_real.h’ and

‘gsl_fft_halfcomplex.h’.
[Function] gsl_fft_real_wavetable * gsl_fft_real_wavetable_alloc

(size t n)

[Function] gsl_fft_halfcomplex_wavetable *
   gsl_fft_halfcomplex_wavetable_alloc (size t n)

These functions prepare trigonometric lookup tables for an FFT of size n real elements. The functions return a pointer to the newly allocated struct if no errors were detected, and a null pointer in the case of error. The length n is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to sin and cos, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then computing the wavetable is a one-off overhead which does not affect the final throughput. The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The appropriate type of wavetable must be used for forward real or inverse half-complex transforms.

[Function] void gsl_fft_real_wavetable_free (gsl fft real wavetable *
   wavetable)

[Function] void gsl_fft_halfcomplex_wavetable_free

(gsl fft halfcomplex wavetable * wavetable) These functions free the memory associated with the wavetable wavetable. The wavetable can be freed if no further FFTs of the same length will be needed.

The mixed radix algorithms require additional working space to hold the intermediate steps of the transform,

[Function] gsl_fft_real_workspace * gsl_fft_real_workspace_alloc

(size t n) This function allocates a workspace for a real transform of length n. The same workspace can be used for both forward real and inverse halfcomplex transforms.

[Function] void gsl_fft_real_workspace_free (gsl fft real workspace *
   workspace)

This function frees the memory associated with the workspace workspace. The workspace can be freed if no further FFTs of the same length will be needed.

The following functions compute the transforms of real and half-complex data,

[Function] int gsl_fft_real_transform (double data[], size t stride, size t n,

const gsl fft real wavetable * wavetable, gsl fft real workspace * work)

[Function] int gsl_fft_halfcomplex_transform (double data[], size t stride,

size t n, const gsl fft halfcomplex wavetable * wavetable,

gsl fft real workspace * work) These functions compute the FFT of data, a real or half-complex array of length n, using a mixed radix decimation-in-frequency algorithm. For gsl_fft_real_transform data is an array of time-ordered real data. For gsl_fft_halfcomplex_transform data contains fourier coefficients in the half-complex ordering described above. There is no restriction on the length n. Efficient modules are provided for subtransforms of length 2, 3, 4 and 5. Any remaining factors are computed with a slow, O(n^{2}), generaln module. The caller must supply a wavetable containing trigonometric lookup tables and a workspace work.

[Function] int gsl_fft_real_unpack (const double real_coefficient[],

gsl complex packed array complex_coefficient[], size t stride, size t n) This function converts a single real array, real coefficient into an equivalent complex array, complex coefficient, (with imaginary part set to zero), suitable for gsl_fft_ complex routines. The algorithm for the conversion is simply,

     for (i = 0; i < n; i++)
       {
         complex_coefficient[i].real
           = real_coefficient[i];
         complex_coefficient[i].imag
           = 0.0;
       }

[Function] int gsl_fft_halfcomplex_unpack (const double

halfcomplex_coefficient[], gsl complex packed array

   complex_coefficient, size t stride, size t n)

This function converts halfcomplex coefficient, an array of half-complex coefficients as returned by gsl_fft_real_transform, into an ordinary complex array, complex coefficient. It fills in the complex array using the symmetry z_{k} = z_{N−k}^{∗} to reconstruct the redundant elements. The algorithm for the conversion is,

     complex_coefficient[0].real
       = halfcomplex_coefficient[0];
     complex_coefficient[0].imag
       = 0.0;
     for (i = 1; i < n - i; i++)
       {
         double hc_real
           = halfcomplex_coefficient[2 * i - 1];
         double hc_imag
           = halfcomplex_coefficient[2 * i];
         complex_coefficient[i].real = hc_real;
         complex_coefficient[i].imag = hc_imag;
         complex_coefficient[n - i].real = hc_real;
         complex_coefficient[n - i].imag = -hc_imag;
       }
     if (i == n - i)
       {
         complex_coefficient[i].real
           = halfcomplex_coefficient[n - 1];
         complex_coefficient[i].imag
           = 0.0;
       }
Here is an example program using gsl_fft_real_transform and gsl_fft_

halfcomplex_inverse. It generates a real signal in the shape of a square pulse. The pulse is fourier transformed to frequency space, and all but the lowest ten frequency components are removed from the array of fourier coefficients returned by gsl_fft_real_transform.

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

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

main (void)
{
int i, n = 100;
double data[n];
gsl_fft_real_wavetable * real;
gsl_fft_halfcomplex_wavetable * hc;
gsl_fft_real_workspace * work;
for (i = 0; i < n; i++)
  {
    data[i] = 0.0;
  }
for (i = n / 3; i < 2 * n / 3; i++)
  {
    data[i] = 1.0;
  }
for (i = 0; i < n; i++)
  {
    printf ("%d: %e\n", i, data[i]);
  }
printf ("\n");
work = gsl_fft_real_workspace_alloc (n);
real = gsl_fft_real_wavetable_alloc (n);
gsl_fft_real_transform (data, 1, n,
                        real, work);
gsl_fft_real_wavetable_free (real);
for (i = 11; i < n; i++)
  {
    data[i] = 0;
  }
hc = gsl_fft_halfcomplex_wavetable_alloc (n);
gsl_fft_halfcomplex_inverse (data, 1, n,
                             hc, work);
gsl_fft_halfcomplex_wavetable_free (hc);
for (i = 0; i < n; i++)
  {
    printf ("%d: %e\n", i, data[i]);

         }
       gsl_fft_real_workspace_free (work);
       return 0;
     }

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

15.8 References and Further Reading

A good starting point for learning more about the FFT is the review article Fast Fourier Transforms: A Tutorial Review and A State of the Art by Duhamel and Vetterli,

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

To find out about the algorithms used in the GSL routines you may want to consult the document GSL FFT Algorithms (it is included in GSL, as ‘doc/fftalgorithms.tex’). This has general information on FFTs and explicit derivations of the implementation for each routine. There are also references to the relevant literature. For convenience some of the more important references are reproduced below.

There are several introductory books on the FFT with example programs, such as The Fast Fourier Transform by Brigham and DFT/FFT and Convolution Algorithms by Burrus and Parks,

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

C. S. Burrus and T. W. Parks. DFT/FFT and Convolution Algorithms. Wiley, 1984.

Both these introductory books cover the radix-2 FFT in some detail. The mixed-radix algorithm at the heart of the fftpack routines is reviewed in Clive Temperton’s paper,

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

The derivation of FFTs for real-valued data is explained in the following two articles, Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C. Sidney Burrus. Real-valued fast fourier transform algorithms. IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-35(6):849–863, 1987. Clive Temperton. Fast mixed-radix real fourier transforms. Journal of Computational Physics, 52:340–350, 1983.

In 1979 the IEEE published a compendium of carefully-reviewed Fortran FFT programs in Programs for Digital Signal Processing. It is a useful reference for implementations of many different FFT algorithms,

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

For large-scale FFT work we recommend the use of the dedicated FFTW library by Frigo and Johnson. The FFTW library is self-optimizing—it automatically tunes itself for each hardware platform in order to achieve maximum performance. It is available under the GNU GPL.

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

The source code for fftpack is available from Netlib,

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

16 Numerical Integration

This chapter describes routines for performing numerical integration (quadrature) of a function in one dimension. There are routines for adaptive and non-adaptive integration of general functions, with specialised routines for specific cases. These include integration over infinite and semi-infinite ranges, singular integrals, including logarithmic singularities, computation of Cauchy principal values and oscillatory integrals. The library reimplements the algorithms used in quadpack, a numerical integration package written by Piessens, Doncker-Kapenga, Uberhuber and Kahaner. Fortran code for quadpack is available on Netlib.

The functions described in this chapter are declared in the header file

‘gsl_integration.h’.

16.1 Introduction

Each algorithm computes an approximation to a definite integral of the form,

I = ∫_{a}^{b} f(x)w(x) dx

where w(x) is a weight function (for general integrands w(x) = 1). The user provides absolute and relative error bounds (epsabs, epsrel ) which specify the following accuracy requirement,

|RESULT − I| ≤ max(epsabs, epsrel |I|)

where RESULT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error ABSERR = |RESULT − I| in such a way that the following inequality holds,

|RESULT − I| ≤ ABSERR ≤ max(epsabs, epsrel |I|)

The routines will fail to converge if the error bounds are too stringent, but always return the best approximation obtained up to that stage.

The algorithms in quadpack use a naming convention based on the following letters, Q - quadrature routine

N - non-adaptive integrator A - adaptive integrator

G - general integrand (user-defined) W - weight function with integrand

S - singularities can be more readily integrated P - points of special difficulty can be supplied I - infinite range of integration O - oscillatory weight function, cos or sin F - Fourier integral C - Cauchy principal value The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule. The higher order rule is used to compute the best approximation to an integral over a small range. The difference between the results of the higher order rule and the lower order rule gives an estimate of the error in the approximation.

16.1.1 Integrands without weight functions

The algorithms for general functions (without a weight function) are based on Gauss-Kronrod rules.

A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of order m. This is extended with additional points between each of the abscissae to give a higher order Kronrod rule of order 2m + 1. The Kronrod rule is efficient because it reuses existing function evaluations from the Gaussian rule.

The higher order Kronrod rule is used as the best approximation to the integral, and the difference between the two rules is used as an estimate of the error in the approximation.

16.1.2 Integrands with weight functions

For integrands with weight functions the algorithms use Clenshaw-Curtis quadrature rules.

A Clenshaw-Curtis rule begins with an n-th order Chebyshev polynomial approximation to the integrand. This polynomial can be integrated exactly to give an approximation to the integral of the original function. The Chebyshev expansion can be extended to higher orders to improve the approximation and provide an estimate of the error.

16.1.3 Integrands with singular weight functions

The presence of singularities (or other behavior) in the integrand can cause slow convergence in the Chebyshev approximation. The modified Clenshaw-Curtis rules used in quadpack separate out several common weight functions which cause slow convergence.

These weight functions are integrated analytically against the Chebyshev polynomials to precompute modified Chebyshev moments. Combining the moments with the Chebyshev approximation to the function gives the desired integral. The use of analytic integration for the singular part of the function allows exact cancellations and substantially improves the overall convergence behavior of the integration.

16.2 QNG non-adaptive Gauss-Kronrod integration

The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions.

[Function] int gsl_integration_qng (const gsl function * f, double a, double b, double epsabs, double epsrel, double * result, double * abserr, size t *

        neval)

This function applies the Gauss-Kronrod 10-point, 21-point, 43-point and 87-point integration rules in succession until an estimate of the integral of f over (a, b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, an estimate of the absolute error, abserr and the number of function evaluations used, neval. The Gauss-Kronrod rules are designed in such a way that each rule uses all the results of its predecessors, in order to minimize the total number of function evaluations.

16.3 QAG adaptive integration

The QAG algorithm is a simple adaptive integration procedure. The integration region is divided into subintervals, and on each iteration the subinterval with the largest estimated error is bisected. This reduces the overall error rapidly, as the subintervals become concentrated around local difficulties in the integrand. These subintervals are managed by a gsl_integration_workspace struct, which handles the memory for the subinterval ranges, results and error estimates.

[Function] gsl_integration_workspace *
   gsl_integration_workspace_alloc (size t n)

This function allocates a workspace sufficient to hold n double precision intervals, their integration results and error estimates.

[Function] void gsl_integration_workspace_free (gsl integration workspace *

   w)

This function frees the memory associated with the workspace w.

[Function] int gsl_integration_qag (const gsl function * f, double a, double b, double epsabs, double epsrel, size t limit, int key, gsl integration workspace * workspace, double * result, double * abserr)

This function applies an integration rule adaptively until an estimate of the integral of f over (a, b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, and an estimate of the absolute error, abserr. The integration rule is determined by the value of key, which should be chosen from the following symbolic names,

     GSL_INTEG_GAUSS15  (key = 1)
     GSL_INTEG_GAUSS21  (key = 2)
     GSL_INTEG_GAUSS31  (key = 3)
     GSL_INTEG_GAUSS41  (key = 4)
     GSL_INTEG_GAUSS51  (key = 5)
     GSL_INTEG_GAUSS61  (key = 6)

corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod rules. The higher-order rules give better accuracy for smooth functions, while lower-order rules save time when the function contains local difficulties, such as discontinuities. On each iteration the adaptive integration strategy bisects the interval with the largest error estimate. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.

16.4 QAGS adaptive integration with singularities

The presence of an integrable singularity in the integration region causes an adaptive routine to concentrate new subintervals around the singularity. As the subintervals decrease in size the successive approximations to the integral converge in a limiting fashion. This approach to the limit can be accelerated using an extrapolation procedure. The QAGS algorithm combines adaptive bisection with the Wynn epsilon-algorithm to speed up the integration of many types of integrable singularities.

[Function] int gsl_integration_qags (const gsl function * f, double a, double b, double epsabs, double epsrel, size t limit, gsl integration workspace *

   workspace, double * result, double * abserr)

This function applies the Gauss-Kronrod 21-point integration rule adaptively until an estimate of the integral of f over (a, b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The results are extrapolated using the epsilon-algorithm, which accelerates the convergence of the integral in the presence of discontinuities and integrable singularities. The function returns the final approximation from the extrapolation, result, and an estimate of the absolute error, abserr. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.

16.5 QAGP adaptive integration with known singular points

[Function] int gsl_integration_qagp (const gsl function * f, double * pts, size t npts, double epsabs, double epsrel, size t limit, gsl integration workspace

   * workspace, double * result, double * abserr)

This function applies the adaptive integration algorithm QAGS taking account of the user-supplied locations of singular points. The array pts of length npts should contain the endpoints of the integration ranges defined by the integration region and locations of the singularities. For example, to integrate over the region (a, b) with break-points at x_{1}, x_{2}, x_{3} (where a < x_{1} < x_{2} < x_{3} < b) the following pts array should be used

     pts[0] = a
     pts[1] = x_1
     pts[2] = x_2
     pts[3] = x_3
     pts[4] = b

with npts = 5. If you know the locations of the singular points in the integration region then this routine will be faster than QAGS.

16.6 QAGI adaptive integration on infinite intervals

[Function] int gsl_integration_qagi (gsl function * f, double epsabs, double

epsrel, size t limit, gsl integration workspace * workspace, double *

   result, double * abserr)

This function computes the integral of the function f over the infinite interval (−∞, +∞). The integral is mapped onto the semi-open interval (0, 1] using the transformation x = (1 − t)/t,

∫_{−∞}^{+∞} dx f(x) = ∫_{0}^{1} dt (f((1 − t)/t) + f(−(1 − t)/t))/t^{2}.

It is then integrated using the QAGS algorithm. The normal 21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the transformation can generate an integrable singularity at the origin. In this case a lower-order rule is more efficient.

[Function] int gsl_integration_qagiu (gsl function * f, double a, double epsabs, double epsrel, size t limit, gsl integration workspace *

   workspace, double * result, double * abserr)

This function computes the integral of the function f over the semi-infinite interval (a, +∞). The integral is mapped onto the semi-open interval (0, 1] using the transformation x = a + (1 − t)/t,

∫_{a}^{+∞} dx f(x) = ∫_{0}^{1} dt f(a + (1 − t)/t)/t^{2}

and then integrated using the QAGS algorithm.

[Function] int gsl_integration_qagil (gsl function * f, double b, double epsabs, double epsrel, size t limit, gsl integration workspace *

   workspace, double * result, double * abserr)

This function computes the integral of the function f over the semi-infinite interval (−∞, b). The integral is mapped onto the semi-open interval (0, 1] using the transformation x = b − (1 − t)/t,

∫_{−∞}^{b} dx f(x) = ∫_{0}^{1} dt f(b − (1 − t)/t)/t^{2}

and then integrated using the QAGS algorithm.

16.7 QAWC adaptive integration for Cauchy principal values

[Function] int gsl_integration_qawc (gsl function * f, double a, double b, double c, double epsabs, double epsrel, size t limit, gsl integration workspace * workspace, double * result, double * abserr)

This function computes the Cauchy principal value of the integral of f over (a, b), with a singularity at c,

I = ∫_{a}^{b} dx \frac{f(x)}{x − c} = lim_{ϵ→0} \{∫_{a}^{c−ϵ} dx \frac{f(x)}{x − c} + ∫_{c+ϵ}^{b} dx \frac{f(x)}{x − c}\}

The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur at the singular point x = c. When a subinterval contains the point x = c or is close to it then a special 25-point modified Clenshaw-Curtis rule is used to control the singularity. Further away from the singularity the algorithm uses an ordinary 15-point Gauss-Kronrod integration rule.

16.8 QAWS adaptive integration for singular functions

The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region. In order to work efficiently the algorithm requires a precomputed table of Chebyshev moments.

[Function] gsl_integration_qaws_table *
   gsl_integration_qaws_table_alloc (double alpha, double beta, int mu,

int nu) This function allocates space for a gsl_integration_qaws_table struct and associated workspace describing a singular weight function W (x) with the parameters (α, β, μ, ν),

W (x) = (x − a)^{α}(b − x)^{β} log^{μ}(x − a) log^{ν}(b − x)

where α > −1, β > −1, and μ = 0, 1, ν = 0, 1. The weight function can take four different forms depending on the values of μ and ν,

W (x) = (x − a)^{α}(b − x)^{β} (μ = 0, ν = 0)
W (x) = (x − a)^{α}(b − x)^{β} log(x − a) (μ = 1, ν = 0)
W (x) = (x − a)^{α}(b − x)^{β} log(b − x) (μ = 0, ν = 1)
W (x) = (x − a)^{α}(b − x)^{β} log(x − a) log(b − x) (μ = 1, ν = 1)

The singular points (a, b) do not have to be specified until the integral is computed, where they are the endpoints of the integration range.

The function returns a pointer to the newly allocated gsl_integration_qaws_table if no errors were detected, and 0 in the case of error.

[Function] int gsl_integration_qaws_table_set (gsl integration qaws table *

t, double alpha, double beta, int mu, int nu) This function modifies the parameters (α, β, μ, ν) of an existing gsl_integration_

qaws_table struct t.
[Function] void gsl_integration_qaws_table_free (gsl integration qaws table

* t) This function frees all the memory associated with the gsl_integration_qaws_table struct t.

[Function] int gsl_integration_qaws (gsl function * f, const double a, const double b, gsl integration qaws table * t, const double epsabs, const double epsrel, const size t limit, gsl integration workspace * workspace, double *

   result, double * abserr)

This function computes the integral of the function f(x) over the interval (a, b) with the singular weight function (x − a)^{α}(b − x)^{β} log^{μ}(x − a) log^{ν}(b − x). The parameters of the weight function (α, β, μ, ν) are taken from the table t. The integral is,

I = ∫_{a}^{b} dx f(x)(x − a)^{α}(b − x)^{β} log^{μ}(x − a) log^{ν}(b − x).

The adaptive bisection algorithm of QAG is used. When a subinterval contains one of the endpoints then a special 25-point modified Clenshaw-Curtis rule is used to control the singularities. For subintervals which do not include the endpoints an ordinary 15-point Gauss-Kronrod integration rule is used.

16.9 QAWO adaptive integration for oscillatory functions

The QAWO algorithm is designed for integrands with an oscillatory factor, sin(ωx) or cos(ωx). In order to work efficiently the algorithm requires a table of Chebyshev moments which must be pre-computed with calls to the functions below.

[Function] gsl_integration_qawo_table *
   gsl_integration_qawo_table_alloc (double omega, double L, enum

gsl integration qawo enum sine, size t n) This function allocates space for a gsl_integration_qawo_table struct and its associated workspace describing a sine or cosine weight function W (x) with the parameters (ω, L),

W (x) = \{ \begin{matrix}sin(ωx) \\ cos(ωx) \end{matrix} \}

The parameter L must be the length of the interval over which the function will be integrated L = b − a. The choice of sine or cosine is made with the parameter sine which should be chosen from one of the two following symbolic values:

     GSL_INTEG_COSINE
     GSL_INTEG_SINE

The gsl_integration_qawo_table is a table of the trigonometric coefficients required in the integration process. The parameter n determines the number of levels of coefficients that are computed. Each level corresponds to one bisection of the interval L, so that n levels are sufficient for subintervals down to the length L/2^{n}. The integration routine gsl_integration_qawo returns the error GSL_ETABLE if the number of levels is insufficient for the requested accuracy.

[Function] int gsl_integration_qawo_table_set (gsl integration qawo table *

t, double omega, double L, enum gsl integration qawo enum sine) This function changes the parameters omega, L and sine of the existing workspace t.

[Function] int gsl_integration_qawo_table_set_length

(gsl integration qawo table * t, double L) This function allows the length parameter L of the workspace t to be changed.

[Function] void gsl_integration_qawo_table_free (gsl integration qawo table

* t) This function frees all the memory associated with the workspace t.

[Function] int gsl_integration_qawo (gsl function * f, const double a, const double epsabs, const double epsrel, const size t limit, gsl integration workspace * workspace, gsl integration qawo table * wf, double * result, double * abserr)

This function uses an adaptive algorithm to compute the integral of f over (a, b) with the weight function sin(ωx) or cos(ωx) defined by the table wf,

I = ∫_{a}^{b} dx f(x) \{ \begin{matrix}sin(ωx) \\ cos(ωx) \end{matrix} \}

The results are extrapolated using the epsilon-algorithm to accelerate the convergence of the integral. The function returns the final approximation from the extrapolation, result, and an estimate of the absolute error, abserr. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace. Those subintervals with “large” widths d where dω > 4 are computed using a 25-point Clenshaw-Curtis integration rule, which handles the oscillatory behavior. Subintervals with a “small” widths where dω < 4 are computed using a 15-point Gauss-Kronrod integration.

16.10 QAWF adaptive integration for Fourier integrals

[Function] int gsl_integration_qawf (gsl function * f, const double a, const double epsabs, const size t limit, gsl integration workspace * workspace, gsl integration workspace * cycle_workspace, gsl integration qawo table * wf, double * result, double * abserr)

This function attempts to compute a Fourier integral of the function f over the semiinfinite interval [a, +∞).

I = ∫_{a}^{+∞} dx f(x) \{ \begin{matrix}sin(ωx) \\ cos(ωx) \end{matrix} \}

The parameter ω and choice of sin or cos is taken from the table wf (the length L can take any value, since it is overridden by this function to a value appropriate for the fourier integration). The integral is computed using the QAWO algorithm over each of the subintervals,

\begin{aligned}C_{1} & = [a, a + c] \\ C_{2} & = [a + c, a + 2c] \\ … & = … \\ C_{k} & = [a + (k − 1)c, a + kc] \end{aligned}

where c = (2 floor(|ω|) + 1)π/|ω|. The width c is chosen to cover an odd number of periods so that the contributions from the intervals alternate in sign and are monotonically decreasing when f is positive and monotonically decreasing. The sum of this sequence of contributions is accelerated using the epsilon-algorithm. This function works to an overall absolute tolerance of abserr. The following strategy is used: on each interval C_{k} the algorithm tries to achieve the tolerance

T OL_{k} = u_{k}abserr

where u_{k} = (1−p)p^{k−1} and p = 9/10. The sum of the geometric series of contributions from each interval gives an overall tolerance of abserr. If the integration of a subinterval leads to difficulties then the accuracy requirement for subsequent intervals is relaxed,

T OL_{k} = u_{k} max(abserr, max_{i<k}{E_{i}})

where E_{k} is the estimated error on the interval C_{k}. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace. The integration over each subinterval uses the memory provided by cycle workspace as workspace for the QAWO algorithm.

16.11 Error codes

In addition to the standard error codes for invalid arguments the functions can return the following values,

GSL_EMAXITER

the maximum number of subdivisions was exceeded.

GSL_EROUND

cannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.

GSL_ESING

a non-integrable singularity or other bad integrand behavior was found in the integration interval.

GSL_EDIVERGE

the integral is divergent, or too slowly convergent to be integrated numerically.

16.12 Examples

The integrator QAGS will handle a large class of definite integrals. For example, consider the following integral, which has a algebraic-logarithmic singularity at the origin,

∫_{0}^{1} x^{−1/2} log(x) dx = −4

The program below computes this integral to a relative accuracy bound of 1e-7.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_integration.h>
double f (double x, void * params) {
  double alpha = *(double *) params;
  double f = log(alpha*x) / sqrt(x);
  return f;
}
int
main (void)
{
  gsl_integration_workspace * w
    = gsl_integration_workspace_alloc (1000);
  double result, error;

double expected = -4.0;
double alpha = 1.0;
gsl_function F;
F.function = &f;
F.params = &alpha;
gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                      w, &result, &error);
printf ("result          = % .18f\n", result);
printf ("exact result    = % .18f\n", expected);
printf ("estimated error = % .18f\n", error);
printf ("actual error    = % .18f\n", result - expected);
printf ("intervals =  %d\n", w->size);
return 0;
}

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

$ ./a.out
result          = -3.999999999999973799
exact result    = -4.000000000000000000
estimated error =  0.000000000000246025
actual error    =  0.000000000000026201
intervals =  8

In fact, the extrapolation procedure used by QAGS produces an accuracy of almost twice as many digits. The error estimate returned by the extrapolation procedure is larger than the actual error, giving a margin of safety of one order of magnitude.

16.13 References and Further Reading

The following book is the definitive reference for quadpack, and was written by the original authors. It provides descriptions of the algorithms, program listings, test programs and examples. It also includes useful advice on numerical integration and many references to the numerical integration literature used in developing quadpack.

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

17 Random Number Generation

The library provides a large collection of random number generators which can be accessed through a uniform interface. Environment variables allow you to select different generators and seeds at runtime, so that you can easily switch between generators without needing to recompile your program. Each instance of a generator keeps track of its own state, allowing the generators to be used in multi-threaded programs. Additional functions are available for transforming uniform random numbers into samples from continuous or discrete probability distributions such as the Gaussian, log-normal or Poisson distributions.

These functions are declared in the header file ‘gsl_rng.h’.

17.1 General comments on random numbers

In 1988, Park and Miller wrote a paper entitled “Random number generators: good ones are hard to find.” [Commun. ACM, 31, 1192–1201]. Fortunately, some excellent random number generators are available, though poor ones are still in common use. You may be happy with the system-supplied random number generator on your computer, but you should be aware that as computers get faster, requirements on random number generators increase. Nowadays, a simulation that calls a random number generator millions of times can often finish before you can make it down the hall to the coffee machine and back.

A very nice review of random number generators was written by Pierre L’Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks, ed. (Wiley, 1997). The chapter is available in postscript from L’Ecuyer’s ftp site (see references). Knuth’s volume on Seminumerical Algorithms (originally published in 1968) devotes 170 pages to random number generators, and has recently been updated in its 3rd edition (1997). It is brilliant, a classic. If you don’t own it, you should stop reading right now, run to the nearest bookstore, and buy it.

A good random number generator will satisfy both theoretical and statistical properties. Theoretical properties are often hard to obtain (they require real math!), but one prefers a random number generator with a long period, low serial correlation, and a tendency not to “fall mainly on the planes.” Statistical tests are performed with numerical simulations. Generally, a random number generator is used to estimate some quantity for which the theory of probability provides an exact answer. Comparison to this exact answer provides a measure of “randomness”.

17.2 The Random Number Generator Interface

It is important to remember that a random number generator is not a “real” function like sine or cosine. Unlike real functions, successive calls to a random number generator yield different return values. Of course that is just what you want for a random number generator, but to achieve this effect, the generator must keep track of some kind of “state” variable. Sometimes this state is just an integer (sometimes just the value of the previously generated random number), but often it is more complicated than that and may involve a whole array of numbers, possibly with some indices thrown in. To use the random number generators, you do not need to know the details of what comprises the state, and besides that varies from algorithm to algorithm.

The random number generator library uses two special structs, gsl_rng_type which holds static information about each type of generator and gsl_rng which describes an instance of a generator created from a given gsl_rng_type.

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

17.3 Random number generator initialization

[Function] gsl_rng * gsl_rng_alloc (const gsl rng type * T)

This function returns a pointer to a newly-created instance of a random number generator of type T. For example, the following code creates an instance of the Tausworthe generator,

     gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);

If there is insufficient memory to create the generator then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM. The generator is automatically initialized with the default seed, gsl_rng_default_ seed. This is zero by default but can be changed either directly or by using the environment variable GSL_RNG_SEED (see Section 17.6 [Random number environment variables], page 170). The details of the available generator types are described later in this chapter.

[Function] void gsl_rng_set (const gsl rng * r, unsigned long int s)

This function initializes (or ‘seeds’) the random number generator. If the generator is seeded with the same value of s on two different runs, the same stream of random numbers will be generated by successive calls to the routines below. If different values of s are supplied, then the generated streams of random numbers should be completely different. If the seed s is zero then the standard seed from the original implementation is used instead. For example, the original Fortran source code for the ranlux generator used a seed of 314159265, and so choosing s equal to zero reproduces

this when using gsl_rng_ranlux.

[Function] void gsl_rng_free (gsl rng * r)

This function frees all the memory associated with the generator r.

17.4 Sampling from a random number generator

The following functions return uniformly distributed random numbers, either as integers or double precision floating point numbers. To obtain non-uniform distributions see Chapter 19 [Random Number Distributions], page 187.

[Function] unsigned long int gsl_rng_get (const gsl rng * r)

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

functions gsl_rng_max (r) and gsl_rng_min (r).

[Function] double gsl_rng_uniform (const gsl rng * r)

This function returns a double precision floating point number uniformly distributed in the range [0,1). The range includes 0.0 but excludes 1.0. The value is typically obtained by dividing the result of gsl_rng_get(r) by gsl_rng_max(r) + 1.0 in double precision. Some generators compute this ratio internally so that they can provide floating point numbers with more than 32 bits of randomness (the maximum number of bits that can be portably represented in a single unsigned long int).

[Function] double gsl_rng_uniform_pos (const gsl rng * r)

This function returns a positive double precision floating point number uniformly distributed in the range (0,1), excluding both 0.0 and 1.0. The number is obtained by sampling the generator with the algorithm of gsl_rng_uniform until a non-zero value is obtained. You can use this function if you need to avoid a singularity at 0.0.

[Function] unsigned long int gsl_rng_uniform_int (const gsl rng * r,

unsigned long int n) This function returns a random integer from 0 to n − 1 inclusive by scaling down and/or discarding samples from the generator r. All integers in the range [0, n − 1] are produced with equal probability. For generators with a non-zero minimum value an offset is applied so that zero is returned with the correct probability.

Note that this function is designed for sampling from ranges smaller than the range of the underlying generator. The parameter n must be less than or equal to the range of the generator r. If n is larger than the range of the generator then the function calls the error handler with an error code of GSL_EINVAL and returns zero.

In particular, this function is not intended for generating the full range of unsigned integer values [0, 2^{32} − 1]. Instead choose a generator with the maximal integer range

and zero mimimum value, such as gsl_rng_ranlxd1, gsl_rng_mt19937 or gsl_rng_

taus, and sample it directly using gsl_rng_get(). The range of each generator can be found using the auxiliary functions described in the next section.

17.5 Auxiliary random number generator functions

The following functions provide information about an existing generator. You should use them in preference to hard-coding the generator parameters into your own code.

[Function] const char * gsl_rng_name (const gsl rng * r)

This function returns a pointer to the name of the generator. For example,

     printf ("r is a '%s' generator\n",
             gsl_rng_name (r));

would print something like r is a 'taus' generator.

[Function] unsigned long int gsl_rng_max (const gsl rng * r)

gsl_rng_max returns the largest value that gsl_rng_get can return.

[Function] unsigned long int gsl_rng_min (const gsl rng * r)

gsl_rng_min returns the smallest value that gsl_rng_get can return. Usually this value is zero. There are some generators with algorithms that cannot return zero, and for these generators the minimum value is 1.

[Function] void * gsl_rng_state (const gsl rng * r)

[Function] size_t gsl_rng_size (const gsl rng * r)

These functions return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream,

     void * state = gsl_rng_state (r);
     size_t n = gsl_rng_size (r);
     fwrite (state, n, 1, stream);

[Function] const gsl_rng_type ** gsl_rng_types_setup (void)

This function returns a pointer to an array of all the available generator types, terminated by a null pointer. The function should be called once at the start of the program, if needed. The following code fragment shows how to iterate over the array of generator types to print the names of the available algorithms,

     const gsl_rng_type **t, **t0;
     t0 = gsl_rng_types_setup ();
     printf ("Available generators:\n");
     for (t = t0; *t != 0; t++)
       {
         printf ("%s\n", (*t)->name);
       }

17.6 Random number environment variables

The library allows you to choose a default generator and seed from the environment variables

GSL_RNG_TYPE and GSL_RNG_SEED and the function gsl_rng_env_setup. This makes it easy

try out different generators and seeds without having to recompile your program.

[Function] const gsl_rng_type * gsl_rng_env_setup (void)

This function reads the environment variables GSL_RNG_TYPE and GSL_RNG_SEED and uses their values to set the corresponding library variables gsl_rng_default and gsl_rng_default_seed. These global variables are defined as follows,

     extern const gsl_rng_type *gsl_rng_default
     extern unsigned long int gsl_rng_default_seed

The environment variable GSL_RNG_TYPE should be the name of a generator, such as taus or mt19937. The environment variable GSL_RNG_SEED should contain the desired seed value. It is converted to an unsigned long int using the C library function strtoul.

If you don’t specify a generator for GSL_RNG_TYPE then gsl_rng_mt19937 is used as the default. The initial value of gsl_rng_default_seed is zero.

Here is a short program which shows how to create a global generator using the envi-

ronment variables GSL_RNG_TYPE and GSL_RNG_SEED,
#include <stdio.h>
#include <gsl/gsl_rng.h>
gsl_rng * r;  /* global generator */
int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);
  printf ("generator type: %s\n", gsl_rng_name (r));
  printf ("seed = %lu\n", gsl_rng_default_seed);
  printf ("first value = %lu\n", gsl_rng_get (r));
  return 0;
}

Running the program without any environment variables uses the initial defaults, an mt19937 generator with a seed of 0,

$ ./a.out
generator type: mt19937
seed = 0
first value = 4293858116

By setting the two variables on the command line we can change the default generator and the seed,

$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
GSL_RNG_TYPE=taus
GSL_RNG_SEED=123
generator type: taus
seed = 123
first value = 2720986350

17.7 Copying random number generator state

The above methods do not expose the random number ‘state’ which changes from call to call. It is often useful to be able to save and restore the state. To permit these practices, a few somewhat more advanced functions are supplied. These include:

[Function] int gsl_rng_memcpy (gsl rng * dest, const gsl rng * src)

This function copies the random number generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.

[Function] gsl_rng * gsl_rng_clone (const gsl rng * r)

This function returns a pointer to a newly created generator which is an exact copy of the generator r.

17.8 Reading and writing random number generator state

The library provides functions for reading and writing the random number state to a file as binary data or formatted text.

[Function] int gsl_rng_fwrite (FILE * stream, const gsl rng * r)

This function writes the random number state of the random number generator r to the stream stream in binary format. The return value is 0 for success and GSL_ EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

[Function] int gsl_rng_fread (FILE * stream, gsl rng * r)

This function reads the random number state into the random number generator r from the open stream stream in binary format. The random number generator r must be preinitialized with the correct random number generator type since type information is not saved. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

17.9 Random number generator algorithms

The functions described above make no reference to the actual algorithm used. This is deliberate so that you can switch algorithms without having to change any of your application source code. The library provides a large number of generators of different types, including simulation quality generators, generators provided for compatibility with other libraries and historical generators from the past.

The following generators are recommended for use in simulation. They have extremely long periods, low correlation and pass most statistical tests.

[Generator] gsl_rng_mt19937

The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a variant of the twisted generalized feedback shift-register algorithm, and is known as the “Mersenne Twister” generator. It has a Mersenne prime period of 2^{19937} − 1 (about 10^{6000}) and is equi-distributed in 623 dimensions. It has passed the diehard statistical tests. It uses 624 words of state per generator and is comparable in speed to the other generators. The original generator used a default seed of 4357 and choosing s equal to zero in gsl_rng_set reproduces this. For more information see,

Makoto Matsumoto and Takuji Nishimura, “Mersenne Twister: A 623dimensionally equidistributed uniform pseudorandom number generator”. ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1 (Jan. 1998), Pages 3–30

The generator gsl_rng_mt19937 uses the second revision of the seeding procedure published by the two authors above in 2002. The original seeding procedures could cause spurious artifacts for some seed values. They are still available through the

alternative generators gsl_rng_mt19937_1999 and gsl_rng_mt19937_1998.
[Generator] gsl_rng_ranlxs0
[Generator] gsl_rng_ranlxs1
[Generator] gsl_rng_ranlxs2

The generator ranlxs0 is a second-generation version of the ranlux algorithm of Lüscher, which produces “luxury random numbers”. This generator provides single precision output (24 bits) at three luxury levels ranlxs0, ranlxs1 and ranlxs2. It uses double-precision floating point arithmetic internally and can be significantly faster than the integer version of ranlux, particularly on 64-bit architectures. The period of the generator is about 10^{171}. The algorithm has mathematically proven properties and can provide truly decorrelated numbers at a known level of randomness. The higher luxury levels provide increased decorrelation between samples as an additional safety margin.

[Generator] gsl_rng_ranlxd1
[Generator] gsl_rng_ranlxd2

These generators produce double precision output (48 bits) from the ranlxs generator. The library provides two luxury levels ranlxd1 and ranlxd2.

[Generator] gsl_rng_ranlux
[Generator] gsl_rng_ranlux389

The ranlux generator is an implementation of the original algorithm developed by Lüscher. It uses a lagged-fibonacci-with-skipping algorithm to produce “luxury random numbers”. It is a 24-bit generator, originally designed for single-precision IEEE floating point numbers. This implementation is based on integer arithmetic, while the second-generation versions ranlxs and ranlxd described above provide floatingpoint implementations which will be faster on many platforms. The period of the generator is about 10^{171}. The algorithm has mathematically proven properties and it can provide truly decorrelated numbers at a known level of randomness. The default level of decorrelation recommended by Lüscher is provided by gsl_rng_ranlux, while gsl_rng_ranlux389 gives the highest level of randomness, with all 24 bits decorrelated. Both types of generator use 24 words of state per generator. For more information see,

M. Lüscher, “A portable high-quality random number generator for lattice field theory calculations”, Computer Physics Communications, 79 (1994) 100–110. F. James, “RANLUX: A Fortran implementation of the high-quality pseudorandom number generator of Lüscher”, Computer Physics Communications, 79 (1994) 111–114

[Generator] gsl_rng_cmrg

This is a combined multiple recursive generator by L’Ecuyer. Its sequence is,

z_{n} = (x_{n} − y_{n}) mod m_{1}

where the two underlying generators x_{n} and y_{n} are,

\begin{aligned}x_{n} & = (a_{1}x_{n−1} + a_{2}x_{n−2} + a_{3}x_{n−3}) mod m_{1} \\ y_{n} & = (b_{1}y_{n−1} + b_{2}y_{n−2} + b_{3}y_{n−3}) mod m_{2} \end{aligned}

with coefficients a_{1} = 0, a_{2} = 63308, a_{3} = −183326, b_{1} = 86098, b_{2} = 0, b_{3} = −539608, and moduli m_{1} = 2^{31} − 1 = 2147483647 and m_{2} = 2145483479.

The period of this generator is lcm(m_{1}^{3}−1, m_{2}^{3}−1), which is approximately 2^{185} (about 10^{56}). It uses 6 words of state per generator. For more information see,

P. L’Ecuyer, “Combined Multiple Recursive Random Number Generators”, Operations Research, 44, 5 (1996), 816–822.

[Generator] gsl_rng_mrg

This is a fifth-order multiple recursive generator by L’Ecuyer, Blouin and Coutre. Its sequence is,

x_{n} = (a_{1}x_{n−1} + a_{5}x_{n−5}) mod m

with a_{1} = 107374182, a_{2} = a_{3} = a_{4} = 0, a_{5} = 104480 and m = 2^{31} − 1.

The period of this generator is about 10^{46}. It uses 5 words of state per generator. More information can be found in the following paper,

P. L’Ecuyer, F. Blouin, and R. Coutre, “A search for good multiple recursive random number generators”, ACM Transactions on Modeling and Computer Simulation 3, 87–98 (1993).

[Generator] gsl_rng_taus
[Generator] gsl_rng_taus2

This is a maximally equidistributed combined Tausworthe generator by L’Ecuyer. The sequence is,

x_{n} = (s_{n}^{1} ⊕ s_{n}^{2} ⊕ s_{n}^{3})

where,

\begin{aligned}s_{n+1}^{1} & = (((s_{n}^{1}&4294967294) ≪ 12) ⊕ (((s_{n}^{1} ≪ 13) ⊕ s_{n}^{1}) ≫ 19)) \\ s_{n+1}^{2} & = (((s_{n}^{2}&4294967288) ≪ 4) ⊕ (((s_{n}^{2} ≪ 2) ⊕ s_{n}^{2}) ≫ 25)) \\ s_{n+1}^{3} & = (((s_{n}^{3}&4294967280) ≪ 17) ⊕ (((s_{n}^{3} ≪ 3) ⊕ s_{n}^{3}) ≫ 11)) \end{aligned}

computed modulo 2^{32}. In the formulas above ⊕ denotes “exclusive-or”. Note that the algorithm relies on the properties of 32-bit unsigned integers and has been implemented using a bitmask of 0xFFFFFFFF to make it work on 64 bit machines.

The period of this generator is 2^{88} (about 10^{26}). It uses 3 words of state per generator. For more information see,

P. L’Ecuyer, “Maximally Equidistributed Combined Tausworthe Generators”, Mathematics of Computation, 65, 213 (1996), 203–213.

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

P. L’Ecuyer, “Tables of Maximally Equidistributed Combined LFSR Generators”, Mathematics of Computation, 68, 225 (1999), 261–269

The generator gsl_rng_taus2 should now be used in preference to gsl_rng_taus.

[Generator] gsl_rng_gfsr4

The gfsr4 generator is like a lagged-fibonacci generator, and produces each number as an xor’d sum of four previous values.

r_{n} = r_{n−A} ⊕ r_{n−B} ⊕ r_{n−C} ⊕ r_{n−D}

Ziff (ref below) notes that “it is now widely known” that two-tap registers (such as R250, which is described below) have serious flaws, the most obvious one being the three-point correlation that comes from the definition of the generator. Nice mathematical properties can be derived for GFSR’s, and numerics bears out the claim that 4-tap GFSR’s with appropriately chosen offsets are as random as can be measured, using the author’s test.

This implementation uses the values suggested the example on p392 of Ziff’s article: A = 471, B = 1586, C = 6988, D = 9689.

If the offsets are appropriately chosen (such as the one ones in this implementation), then the sequence is said to be maximal; that means that the period is 2^{D} − 1, where D is the longest lag. (It is one less than 2^{D} because it is not permitted to have all zeros in the ra[] array.) For this implementation with D = 9689 that works out to about 10^{2917}.

Note that the implementation of this generator using a 32-bit integer amounts to 32 parallel implementations of one-bit generators. One consequence of this is that the period of this 32-bit generator is the same as for the one-bit generator. Moreover, this independence means that all 32-bit patterns are equally likely, and in particular that 0 is an allowed random value. (We are grateful to Heiko Bauke for clarifying for us these properties of GFSR random number generators.)

For more information see,

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

17.10 Unix random number generators

The standard Unix random number generators rand, random and rand48 are provided as part of GSL. Although these generators are widely available individually often they aren’t all available on the same platform. This makes it difficult to write portable code using them and so we have included the complete set of Unix generators in GSL for convenience. Note that these generators don’t produce high-quality randomness and aren’t suitable for work requiring accurate statistics. However, if you won’t be measuring statistical quantities and just want to introduce some variation into your program then these generators are quite acceptable.

[Generator] gsl_rng_rand

This is the BSD rand() generator. Its sequence is

x_{n+1} = (ax_{n} + c) mod m

with a = 1103515245, c = 12345 and m = 2^{31}. The seed specifies the initial value, x_{1}. The period of this generator is 2^{31}, and it uses 1 word of storage per generator.

[Generator] gsl_rng_random_bsd
[Generator] gsl_rng_random_libc5
[Generator] gsl_rng_random_glibc2

These generators implement the random() family of functions, a set of linear feedback shift register generators originally used in BSD Unix. There are several versions of random() in use today: the original BSD version (e.g. on SunOS4), a libc5 version (found on older GNU/Linux systems) and a glibc2 version. Each version uses a different seeding procedure, and thus produces different sequences.

The original BSD routines accepted a variable length buffer for the generator state, with longer buffers providing higher-quality randomness. The random() function implemented algorithms for buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm with the largest length that would fit into the user-supplied buffer was used. To support these algorithms additional generators are available with the following names,

     gsl_rng_random8_bsd
     gsl_rng_random32_bsd
     gsl_rng_random64_bsd
     gsl_rng_random128_bsd
     gsl_rng_random256_bsd

where the numeric suffix indicates the buffer length. The original BSD random function used a 128-byte default buffer and so gsl_rng_random_bsd has been made equivalent to gsl_rng_random128_bsd. Corresponding versions of the libc5 and glibc2 generators are also available, with the names gsl_rng_random8_libc5, gsl_rng_

random8_glibc2, etc.
[Generator] gsl_rng_rand48

This is the Unix rand48 generator. Its sequence is

x_{n+1} = (ax_{n} + c) mod m

defined on 48-bit unsigned integers with a = 25214903917, c = 11 and m = 2^{48}. The seed specifies the upper 32 bits of the initial value, x_{1}, with the lower 16 bits set to 0x330E. The function gsl_rng_get returns the upper 32 bits from each term of the sequence. This does not have a direct parallel in the original rand48 functions, but forcing the result to type long int reproduces the output of mrand48. The function gsl_rng_uniform uses the full 48 bits of internal state to return the double precision number x_{n}/m, which is equivalent to the function drand48. Note that some versions of the GNU C Library contained a bug in mrand48 function which caused it to produce different results (only the lower 16-bits of the return value were set).

17.11 Other random number generators

The generators in this section are provided for compatibility with existing libraries. If you are converting an existing program to use GSL then you can select these generators to check your new implementation against the original one, using the same random number generator. After verifying that your new program reproduces the original results you can then switch to a higher-quality generator.

Note that most of the generators in this section are based on single linear congruence relations, which are the least sophisticated type of generator. In particular, linear congruences have poor properties when used with a non-prime modulus, as several of these routines do (e.g. with a power of two modulus, 2^{31} or 2^{32}). This leads to periodicity in the least significant bits of each number, with only the higher bits having any randomness. Thus if you want to produce a random bitstream it is best to avoid using the least significant bits.

[Generator] gsl_rng_ranf

This is the CRAY random number generator RANF. Its sequence is

x_{n+1} = (ax_{n}) mod m

defined on 48-bit unsigned integers with a = 44485709377909 and m = 2^{48}. The seed specifies the lower 32 bits of the initial value, x_{1}, with the lowest bit set to prevent the seed taking an even value. The upper 16 bits of x_{1} are set to 0. A consequence of this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc produce the same sequences. The generator compatible with the CRAY MATHLIB routine RANF. It produces double precision floating point numbers which should be identical to those from the original RANF. There is a subtlety in the implementation of the seeding. The initial state is reversed through one step, by multiplying by the modular inverse of a mod m. This is done for compatibility with the original CRAY implementation. Note that you can only seed the generator with integers up to 2^{32}, while the original CRAY implementation uses non-portable wide integers which can cover all 2^{48} states of the generator. The function gsl_rng_get returns the upper 32 bits from each term of the sequence. The function gsl_rng_uniform uses the full 48 bits to return the double precision number x_{n}/m. The period of this generator is 2^{46}.

[Generator] gsl_rng_ranmar

This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman and Tsang. It is a 24-bit generator, originally designed for single-precision IEEE floating point numbers. It was included in the CERNLIB high-energy physics library.

[Generator] gsl_rng_r250

This is the shift-register generator of Kirkpatrick and Stoll. The sequence is based on the recurrence

x_{n} = x_{n−103} ⊕ x_{n−250}

where ⊕ denotes “exclusive-or”, defined on 32-bit words. The period of this generator is about 2^{250} and it uses 250 words of state per generator. For more information see,

S. Kirkpatrick and E. Stoll, “A very fast shift-register sequence random number generator”, Journal of Computational Physics, 40, 517–526 (1981)

[Generator] gsl_rng_tt800

This is an earlier version of the twisted generalized feedback shift-register generator, and has been superseded by the development of MT19937. However, it is still an acceptable generator in its own right. It has a period of 2^{800} and uses 33 words of storage per generator.

For more information see,

Makoto Matsumoto and Yoshiharu Kurita, “Twisted GFSR Generators II”, ACM Transactions on Modelling and Computer Simulation, Vol. 4, No. 3, 1994, pages 254–266.

[Generator] gsl_rng_vax

This is the VAX generator MTH$RANDOM. Its sequence is,

x_{n+1} = (ax_{n} + c) mod m

with a = 69069, c = 1 and m = 2^{32}. The seed specifies the initial value, x_{1}. The period of this generator is 2^{32} and it uses 1 word of storage per generator.

[Generator] gsl_rng_transputer

This is the random number generator from the INMOS Transputer Development system. Its sequence is,

x_{n+1} = (ax_{n}) mod m

with a = 1664525 and m = 2^{32}. The seed specifies the initial value, x_{1}.

[Generator] gsl_rng_randu

This is the IBM RANDU generator. Its sequence is

x_{n+1} = (ax_{n}) mod m

with a = 65539 and m = 2^{31}. The seed specifies the initial value, x_{1}. The period of this generator was only 2^{29}. It has become a textbook example of a poor generator.

[Generator] gsl_rng_minstd

This is Park and Miller’s “minimal standard” minstd generator, a simple linear congruence which takes care to avoid the major pitfalls of such algorithms. Its sequence is,

x_{n+1} = (ax_{n}) mod m

with a = 16807 and m = 2^{31} − 1 = 2147483647. The seed specifies the initial value, x_{1}. The period of this generator is about 2^{31}.

This generator is used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function). It is also sometimes known by the acronym “GGL” (I’m not sure what that stands for).

For more information see,

Park and Miller, “Random Number Generators: Good ones are hard to find”, Communications of the ACM, October 1988, Volume 31, No 10, pages 1192–1201.

[Generator] gsl_rng_uni
[Generator] gsl_rng_uni32

This is a reimplementation of the 16-bit SLATEC random number generator RUNIF. A generalization of the generator to 32 bits is provided by gsl_rng_uni32. The original source code is available from NETLIB.

[Generator] gsl_rng_slatec

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

[Generator] gsl_rng_zuf

This is the ZUFALL lagged Fibonacci series generator of Peterson. Its sequence is,

\begin{aligned}t & = u_{n−273} + u_{n−607} \\ u_{n} & = t − floor(t) \end{aligned}

The original source code is available from NETLIB. For more information see,

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

[Generator] gsl_rng_borosh13

This is the Borosh-Niederreiter random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., pages 106–108. Its sequence is,

x_{n+1} = (ax_{n}) mod m

with a = 1812433253 and m = 2^{32}. The seed specifies the initial value, x_{1}.

[Generator] gsl_rng_coveyou

This is the Coveyou random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., Section 3.2.2. Its sequence is,

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

with m = 2^{32}. The seed specifies the initial value, x_{1}.

[Generator] gsl_rng_fishman18

This is the Fishman, Moore III random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., pages 106–108. Its sequence is,

x_{n+1} = (ax_{n}) mod m

with a = 62089911 and m = 2^{31} − 1. The seed specifies the initial value, x_{1}.

[Generator] gsl_rng_fishman20

This is the Fishman random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,

x_{n+1} = (ax_{n}) mod m

with a = 48271 and m = 2^{31} − 1. The seed specifies the initial value, x_{1}.

[Generator] gsl_rng_fishman2x

This is the L’Ecuyer–Fishman random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,

z_{n+1} = (x_{n} − y_{n}) mod m

with m = 2^{31} − 1. x_{n} and y_{n} are given by the fishman20 and lecuyer21 algorithms. The seed specifies the initial value, x_{1}.

[Generator] gsl_rng_knuthran2

This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,

x_{n} = (a_{1}x_{n−1} + a_{2}x_{n−2}) mod m

with a_{1} = 271828183, a_{2} = 314159269, and m = 2^{31} − 1.

[Generator] gsl_rng_knuthran

This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6. Knuth provides its C code.

[Generator] gsl_rng_lecuyer21

This is the L’Ecuyer random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 106–108. Its sequence is,

x_{n+1} = (ax_{n}) mod m

with a = 40692 and m = 2^{31} − 249. The seed specifies the initial value, x_{1}.

[Generator] gsl_rng_waterman14

This is the Waterman random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 106–108. Its sequence is,

x_{n+1} = (ax_{n}) mod m

with a = 1566083941 and m = 2^{32}. The seed specifies the initial value, x_{1}.

17.12 Performance

The following table shows the relative performance of a selection the available random number generators. The fastest simulation quality generators are taus, gfsr4 and mt19937. The generators which offer the best mathematically-proven quality are those based on the ranlux algorithm.

1754 k ints/sec,    870 k doubles/sec, taus
1613 k ints/sec,    855 k doubles/sec, gfsr4
1370 k ints/sec,    769 k doubles/sec, mt19937
 565 k ints/sec,    571 k doubles/sec, ranlxs0
 400 k ints/sec,    405 k doubles/sec, ranlxs1
 490 k ints/sec,    389 k doubles/sec, mrg
 407 k ints/sec,    297 k doubles/sec, ranlux
 243 k ints/sec,    254 k doubles/sec, ranlxd1
 251 k ints/sec,    253 k doubles/sec, ranlxs2

 238 k ints/sec,    215 k doubles/sec, cmrg
 247 k ints/sec,    198 k doubles/sec, ranlux389
 141 k ints/sec,    140 k doubles/sec, ranlxd2
1852 k ints/sec,    935 k doubles/sec, ran3
 813 k ints/sec,    575 k doubles/sec, ran0
 787 k ints/sec,    476 k doubles/sec, ran1
 379 k ints/sec,    292 k doubles/sec, ran2

17.13 Examples

The following program demonstrates the use of a random number generator to produce uniform random numbers in the range [0.0, 1.0),

#include <stdio.h>
#include <gsl/gsl_rng.h>
int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;
  int i, n = 10;
  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);
  for (i = 0; i < n; i++)
    {
      double u = gsl_rng_uniform (r);
      printf ("%.5f\n", u);
    }
  gsl_rng_free (r);
  return 0;
}

Here is the output of the program,

$ ./a.out
0.99974
0.16291
0.28262
0.94720
0.23166
0.48497

0.95748
0.74431
0.54004
0.73995

The numbers depend on the seed used by the generator. The default seed can be changed with the GSL_RNG_SEED environment variable to produce a different stream of numbers. The generator itself can be changed using the environment variable GSL_RNG_TYPE. Here is the output of the program using a seed value of 123 and the multiple-recursive generator

mrg,
$ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out
GSL_RNG_TYPE=mrg
GSL_RNG_SEED=123
0.33050
0.86631
0.32982
0.67620
0.53391
0.06457
0.16847
0.70229
0.04371
0.86374

17.14 References and Further Reading

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

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

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

P. L’Ecuyer, “Random Number Generation”, Chapter 4 of the Handbook on Simulation, Jerry Banks Ed., Wiley, 1998, 93–137.

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

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

DIEHARD source code G. Marsaglia,

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

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

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

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

17.15 Acknowledgements

Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for making the source code to their generators (MT19937, MM&TN; TT800, MM&YK) available under the GNU General Public License. Thanks to Martin Lüscher for providing notes and source code for the ranlxs and ranlxd generators.

18 Quasi-Random Sequences

This chapter describes functions for generating quasi-random sequences in arbitrary dimensions. A quasi-random sequence progressively covers a d-dimensional space with a set of points that are uniformly distributed. Quasi-random sequences are also known as lowdiscrepancy sequences. The quasi-random sequence generators use an interface that is similar to the interface for random number generators, except that seeding is not required—each generator produces a single sequence.

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

18.1 Quasi-random number generator initialization

[Function] gsl_qrng * gsl_qrng_alloc (const gsl qrng type * T, unsigned int d)

This function returns a pointer to a newly-created instance of a quasi-random sequence generator of type T and dimension d. If there is insufficient memory to create the generator then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] void gsl_qrng_free (gsl qrng * q)

This function frees all the memory associated with the generator q.

[Function] void gsl_qrng_init (gsl qrng * q)

This function reinitializes the generator q to its starting point. Note that quasirandom sequences do not use a seed and always produce the same set of values.

18.2 Sampling from a quasi-random number generator

[Function] int gsl_qrng_get (const gsl qrng * q, double x[])

This function stores the next point from the sequence generator q in the array x. The space available for x must match the dimension of the generator. The point x will lie in the range 0 < x_{i} < 1 for each x_{i}.

18.3 Auxiliary quasi-random number generator functions

[Function] const char * gsl_qrng_name (const gsl qrng * q)

This function returns a pointer to the name of the generator.

[Function] size_t gsl_qrng_size (const gsl qrng * q)

[Function] void * gsl_qrng_state (const gsl qrng * q)

These functions return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream,

     void * state = gsl_qrng_state (q);
     size_t n = gsl_qrng_size (q);
     fwrite (state, n, 1, stream);

18.4 Saving and resorting quasi-random number generator state

[Function] int gsl_qrng_memcpy (gsl qrng * dest, const gsl qrng * src)

This function copies the quasi-random sequence generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.

[Function] gsl_qrng * gsl_qrng_clone (const gsl qrng * q)

This function returns a pointer to a newly created generator which is an exact copy of the generator q.

18.5 Quasi-random number generator algorithms

The following quasi-random sequence algorithms are available,

[Generator] gsl_qrng_niederreiter_2

This generator uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim. 2, 195 (1992). It is valid up to 12 dimensions.

[Generator] gsl_qrng_sobol

This generator uses the Sobol sequence described in Antonov, Saleev, USSR Comput. Maths. Math. Phys. 19, 252 (1980). It is valid up to 40 dimensions.

18.6 Examples

The following program prints the first 1024 points of the 2-dimensional Sobol sequence.

#include <stdio.h>
#include <gsl/gsl_qrng.h>
int
main (void)
{
  int i;
  gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);
  for (i = 0; i < 1024; i++)
    {
      double v[2];
      gsl_qrng_get (q, v);
      printf ("%.5f %.5f\n", v[0], v[1]);
    }
  gsl_qrng_free (q);
  return 0;
}

Here is the output from the program,

$ ./a.out
0.50000 0.50000

0.75000 0.25000
0.25000 0.75000
0.37500 0.37500
0.87500 0.87500
0.62500 0.12500
0.12500 0.62500
....

It can be seen that successive points progressively fill-in the spaces between previous points.

The following plot shows the distribution in the x-y plane of the first 1024 points from the Sobol sequence,

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

18.7 References

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

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

19 Random Number Distributions

This chapter describes functions for generating random variates and computing their probability distributions. Samples from the distributions described in this chapter can be obtained using any of the random number generators in the library as an underlying source of randomness.

In the simplest cases a non-uniform distribution can be obtained analytically from the uniform distribution of a random number generator by applying an appropriate transformation. This method uses one call to the random number generator. More complicated distributions are created by the acceptance-rejection method, which compares the desired distribution against a distribution which is similar and known analytically. This usually requires several samples from the generator.

The library also provides cumulative distribution functions and inverse cumulative distribution functions, sometimes referred to as quantile functions. The cumulative distribution functions and their inverses are computed separately for the upper and lower tails of the distribution, allowing full accuracy to be retained for small results.

The functions for random variates and probability density functions described in this section are declared in ‘gsl_randist.h’. The corresponding cumulative distribution functions are declared in ‘gsl_cdf.h’.

Note that the discrete random variate functions always return a value of type unsigned int, and on most platforms this has a maximum value of 2^{32} − 1 ≈ 4.29 × 10^{9}. They should only be called with a safe range of parameters (where there is a negligible probability of a variate exceeding this limit) to prevent incorrect results due to overflow.

19.1 Introduction

Continuous random number distributions are defined by a probability density function, p(x), such that the probability of x occurring in the infinitesimal range x to x + dx is p dx.

The cumulative distribution function for the lower tail P (x) is defined by the integral,

P (x) = ∫_{−∞}^{x} dx′p(x′)

and gives the probability of a variate taking a value less than x.

The cumulative distribution function for the upper tail Q(x) is defined by the integral,

Q(x) = ∫_{x}^{+∞} dx′p(x′)

and gives the probability of a variate taking a value greater than x.

The upper and lower cumulative distribution functions are related by P (x) + Q(x) = 1 and satisfy 0 ≤ P (x) ≤ 1, 0 ≤ Q(x) ≤ 1.

The inverse cumulative distributions, x = P^{−1}(P) and x = Q^{−1}(Q) give the values of x which correspond to a specific value of P or Q. They can be used to find confidence limits from probability values.

For discrete distributions the probability of sampling the integer value k is given by p(k), where ∑_{k} p(k) = 1. The cumulative distribution for the lower tail P (k) of a discrete distribution is defined as,

P (k) = ∑_{i≤k} p(i)

where the sum is over the allowed range of the distribution less than or equal to k.

The cumulative distribution for the upper tail of a discrete distribution Q(k) is defined as

Q(k) = ∑_{i>k} p(i)

giving the sum of probabilities for all values greater than k. These two definitions satisfy the identity P (k) + Q(k) = 1.

If the range of the distribution is 1 to n inclusive then P (n) = 1, Q(n) = 0 while P (1) = p(1), Q(1) = 1 − p(1).

19.2 The Gaussian Distribution

[Function] double gsl_ran_gaussian (const gsl rng * r, double sigma)

This function returns a Gaussian random variate, with mean zero and standard deviation sigma. The probability distribution for Gaussian random variates is,

p(x)dx = \frac{1}{\sqrt{2πσ^{2}}} exp(−x^{2}/2σ^{2})dx

for x in the range −∞ to +∞. Use the transformation z = μ + x on the numbers returned by gsl_ran_gaussian to obtain a Gaussian distribution with mean μ. This function uses the Box-Mueller algorithm which requires two calls to the random number generator r.

[Function] double gsl_ran_gaussian_pdf (double x, double sigma)

This function computes the probability density p(x) at x for a Gaussian distribution with standard deviation sigma, using the formula given above.

σ = 2 σ = 1 Gaussian Distribution x p(x)  5 4 3 2 1 0 -1 -2 -3 -4 -5 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_ran_gaussian_ziggurat (const gsl rng * r, double
   sigma)
[Function] double gsl_ran_gaussian_ratio_method (const gsl rng * r, double
   sigma)

This function computes a Gaussian random variate using the alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva ratio methods. The Ziggurat algorithm is the fastest available algorithm in most cases.

[Function] double gsl_ran_ugaussian (const gsl rng * r)
[Function] double gsl_ran_ugaussian_pdf (double x)

[Function] double gsl_ran_ugaussian_ratio_method (const gsl rng * r)

These functions compute results for the unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.

[Function] double gsl_cdf_gaussian_P (double x, double sigma)
[Function] double gsl_cdf_gaussian_Q (double x, double sigma)
[Function] double gsl_cdf_gaussian_Pinv (double P, double sigma)
[Function] double gsl_cdf_gaussian_Qinv (double Q, double sigma)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Gaussian distribution with standard deviation sigma.

[Function] double gsl_cdf_ugaussian_P (double x)
[Function] double gsl_cdf_ugaussian_Q (double x)
[Function] double gsl_cdf_ugaussian_Pinv (double P)
[Function] double gsl_cdf_ugaussian_Qinv (double Q)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the unit Gaussian distribution.

19.3 The Gaussian Tail Distribution

[Function] double gsl_ran_gaussian_tail (const gsl rng * r, double a, double

   sigma)

This function provides random variates from the upper tail of a Gaussian distribution with standard deviation sigma. The values returned are larger than the lower limit a, which must be positive. The method is based on Marsaglia’s famous rectangle-wedge-tail algorithm (Ann. Math. Stat. 32, 894–899 (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586 (exercise 11). The probability distribution for Gaussian tail random variates is,

p(x)dx = \frac{1}{N(a; σ)\sqrt{2πσ^{2}}} exp(−x^{2}/2σ^{2})dx

for x > a where N(a; σ) is the normalization constant,

N(a; σ) = \frac{1}{2}erfc (\frac{a}{\sqrt{2σ^{2}}} ) .

[Function] double gsl_ran_gaussian_tail_pdf (double x, double a, double
   sigma)

This function computes the probability density p(x) at x for a Gaussian tail distribution with standard deviation sigma and lower limit a, using the formula given above.

σ = 1, a = 1.5 Gaussian Tail Distribution x p(x)  5 4 3 2 1 0 2 1.5 1 0.5 0

[Function] double gsl_ran_ugaussian_tail (const gsl rng * r, double a)

[Function] double gsl_ran_ugaussian_tail_pdf (double x, double a)

These functions compute results for the tail of a unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.

19.4 The Bivariate Gaussian Distribution

[Function] void gsl_ran_bivariate_gaussian (const gsl rng * r, double

sigma_x, double sigma_y, double rho, double * x, double * y) This function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient rho and standard deviations sigma x and sigma y in the x and y directions. The probability distribution for bivariate Gaussian random variates is,

p(x, y)dxdy = \frac{1}{2πσ_{x}σ_{y}\sqrt{1 − ρ^{2}}} exp (−\frac{(x^{2}/σ_{x}^{2} + y^{2}/σ_{y}^{2} − 2ρxy/(σ_{x}σ_{y}))}{2(1 − ρ^{2})}) dxdy

for x, y in the range −∞ to +∞. The correlation coefficient rho should lie between 1 and −1.

[Function] double gsl_ran_bivariate_gaussian_pdf (double x, double y,

double sigma_x, double sigma_y, double rho) This function computes the probability density p(x, y) at (x,y) for a bivariate Gaussian distribution with standard deviations sigma x, sigma y and correlation coefficient rho, using the formula given above.

y x σ_{x} = 1, σ_{y} = 1, ρ = 0.9 Bivariate Gaussian Distribution 2 1 0 -1 -2 2 1 0 -1 -2

19.5 The Exponential Distribution

[Function] double gsl_ran_exponential (const gsl rng * r, double mu)

This function returns a random variate from the exponential distribution with mean mu. The distribution is,

p(x)dx = \frac{1}{μ} exp(−x/μ)dx

for x ≥ 0.

[Function] double gsl_ran_exponential_pdf (double x, double mu)

This function computes the probability density p(x) at x for an exponential distribution with mean mu, using the formula given above.

μ = 2 μ = 1 Exponential Distribution x p(x)  3 2 1 0 1 0.5 0

[Function] double gsl_cdf_exponential_P (double x, double mu)
[Function] double gsl_cdf_exponential_Q (double x, double mu)
[Function] double gsl_cdf_exponential_Pinv (double P, double mu)
[Function] double gsl_cdf_exponential_Qinv (double Q, double mu)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the exponential distribution with mean mu.

19.6 The Laplace Distribution

[Function] double gsl_ran_laplace (const gsl rng * r, double a)

This function returns a random variate from the Laplace distribution with width a. The distribution is,

p(x)dx = \frac{1}{2a} exp(−|x/a|)dx

for −∞ < x < ∞.

[Function] double gsl_ran_laplace_pdf (double x, double a)

This function computes the probability density p(x) at x for a Laplace distribution with width a, using the formula given above.

a = 2 a = 1 Laplace Distribution (Two-sided Exponential) x p(x)  5 4 3 2 1 0 -1 -2 -3 -4 -5 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_laplace_P (double x, double a)
[Function] double gsl_cdf_laplace_Q (double x, double a)
[Function] double gsl_cdf_laplace_Pinv (double P, double a)
[Function] double gsl_cdf_laplace_Qinv (double Q, double a)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Laplace distribution with width a.

19.7 The Exponential Power Distribution

[Function] double gsl_ran_exppow (const gsl rng * r, double a, double b)

This function returns a random variate from the exponential power distribution with scale parameter a and exponent b. The distribution is,

p(x)dx = \frac{1}{2aΓ(1 + 1/b)} exp(−|x/a|^{b})dx

for x ≥ 0. For b = 1 this reduces to the Laplace distribution. For b = 2 it has the same form as a gaussian distribution, but with a = \sqrt{2}σ.

[Function] double gsl_ran_exppow_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for an exponential power distribution with scale parameter a and exponent b, using the formula given above.

a = 1, b = 0.5 a = 1, b = 2.5 Exponential Power Distribution x p(x)  5 4 3 2 1 0 -1 -2 -3 -4 -5 0.8 0.6 0.4 0.2 0

[Function] double gsl_cdf_exppow_P (double x, double a, double b)

[Function] double gsl_cdf_exppow_Q (double x, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) for the exponential power distribution with parameters a and b.

19.8 The Cauchy Distribution

[Function] double gsl_ran_cauchy (const gsl rng * r, double a)

This function returns a random variate from the Cauchy distribution with scale parameter a. The probability distribution for Cauchy random variates is,

p(x)dx = \frac{1}{aπ(1 + (x/a)^{2})}dx

for x in the range −∞ to +∞. The Cauchy distribution is also known as the Lorentz distribution.

[Function] double gsl_ran_cauchy_pdf (double x, double a)

This function computes the probability density p(x) at x for a Cauchy distribution with scale parameter a, using the formula given above.

a = 2 a = 1 Cauchy Distribution x p(x)  5 4 3 2 1 0 -1 -2 -3 -4 -5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_cauchy_P (double x, double a)

[Function] double gsl_cdf_cauchy_Q (double x, double a)

[Function] double gsl_cdf_cauchy_Pinv (double P, double a)
[Function] double gsl_cdf_cauchy_Qinv (double Q, double a)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Cauchy distribution with scale parameter a.

19.9 The Rayleigh Distribution

[Function] double gsl_ran_rayleigh (const gsl rng * r, double sigma)

This function returns a random variate from the Rayleigh distribution with scale parameter sigma. The distribution is,

p(x)dx = \frac{x}{σ^{2}} exp(−x^{2}/(2σ^{2}))dx

for x > 0.

[Function] double gsl_ran_rayleigh_pdf (double x, double sigma)

This function computes the probability density p(x) at x for a Rayleigh distribution with scale parameter sigma, using the formula given above.

σ = 2 σ = 1 Rayleigh Distribution x p(x)  5 4 3 2 1 0 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_rayleigh_P (double x, double sigma)
[Function] double gsl_cdf_rayleigh_Q (double x, double sigma)
[Function] double gsl_cdf_rayleigh_Pinv (double P, double sigma)
[Function] double gsl_cdf_rayleigh_Qinv (double Q, double sigma)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Rayleigh distribution with scale parameter sigma.

19.10 The Rayleigh Tail Distribution

[Function] double gsl_ran_rayleigh_tail (const gsl rng * r, double a, double

   sigma)

This function returns a random variate from the tail of the Rayleigh distribution with scale parameter sigma and a lower limit of a. The distribution is,

p(x)dx = \frac{x}{σ^{2}} exp((a^{2} − x^{2})/(2σ^{2}))dx

for x > a.

[Function] double gsl_ran_rayleigh_tail_pdf (double x, double a, double
   sigma)

This function computes the probability density p(x) at x for a Rayleigh tail distribution with scale parameter sigma and lower limit a, using the formula given above.

a = 0.5, σ = 2 a = 1, σ = 1 Rayleigh Tail Distribution x p(x)  5 4 3 2 1 0 1 0.5 0

19.11 The Landau Distribution

[Function] double gsl_ran_landau (const gsl rng * r)

This function returns a random variate from the Landau distribution. The probability distribution for Landau random variates is defined analytically by the complex integral,

p(x) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} ds exp(s log(s) + xs)

For numerical purposes it is more convenient to use the following equivalent form of the integral,

p(x) = (1/π) ∫_{0}^{∞} dt exp(−t log(t) − xt) sin(πt).

[Function] double gsl_ran_landau_pdf (double x)

This function computes the probability density p(x) at x for the Landau distribution using an approximation to the formula given above.

Landau Distribution x p(x)  10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 0.2 0.1 0

19.12 The Levy alpha-Stable Distributions

[Function] double gsl_ran_levy (const gsl rng * r, double c, double alpha)

This function returns a random variate from the Levy symmetric stable distribution with scale c and exponent alpha. The symmetric stable probability distribution is defined by a fourier transform,

p(x) = \frac{1}{2π} ∫_{−∞}^{+∞} dt exp(−itx − |ct|^{α})

There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function. For α = 1 the distribution reduces to the Cauchy distribution. For α = 2 it is a Gaussian distribution with σ = \sqrt{2}c. For α < 1 the tails of the distribution become extremely wide. The algorithm only works for 0 < α ≤ 2.

c = 1, α = 2.0 c = 1, α = 1.0 Levy Distribution x p(x)  5 4 3 2 1 0 -1 -2 -3 -4 -5 0.4 0.3 0.2 0.1 0

19.13 The Levy skew alpha-Stable Distribution

[Function] double gsl_ran_levy_skew (const gsl rng * r, double c, double alpha,

double beta) This function returns a random variate from the Levy skew stable distribution with scale c, exponent alpha and skewness parameter beta. The skewness parameter must lie in the range [−1, 1]. The Levy skew stable probability distribution is defined by a fourier transform,

p(x) = \frac{1}{2π} ∫_{−∞}^{+∞} dt exp(−itx − |ct|^{α}(1 − iβsign(t) tan(πα/2)))

When α = 1 the term tan(πα/2) is replaced by −(2/π) log |t|. There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function. For α = 2 the distribution reduces to a Gaussian distribution with σ = \sqrt{2}c and the skewness parameter has no effect. For α < 1 the tails of the distribution become extremely wide. The symmetric distribution corresponds to β = 0. The algorithm only works for 0 < α ≤ 2.

The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, α, β) then the sum Y = X_{1} + X_{2} + … + X_{N} will also be distributed as an alpha-stable variate, p(N^{1/α}c, α, β).

c = 1, α = 1.0, β = 1.0 Levy Skew Distribution x p(x)  5 4 3 2 1 0 -1 -2 -3 -4 -5 0.05 0

19.14 The Gamma Distribution

[Function] double gsl_ran_gamma (const gsl rng * r, double a, double b)

This function returns a random variate from the gamma distribution. The distribution function is,

p(x)dx = \frac{1}{Γ(a)b^{a}} x^{a−1}e^{−x/b}dx

for x > 0. The gamma distribution with an integer parameter a is known as the Erlang distribution. The variates are computed using the algorithms from Knuth (vol 2).

[Function] double gsl_ran_gamma_mt (const gsl rng * r, double a, double b)

This function returns a gamma variate using the Marsaglia-Tsang fast gamma method.

[Function] double gsl_ran_gamma_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for a gamma distribution with parameters a and b, using the formula given above.

a = 3 a = 2 a = 1 Gamma Distribution x p(x)  5 4 3 2 1 0 1 0.5 0

[Function] double gsl_cdf_gamma_P (double x, double a, double b)

[Function] double gsl_cdf_gamma_Q (double x, double a, double b)

[Function] double gsl_cdf_gamma_Pinv (double P, double a, double b)

[Function] double gsl_cdf_gamma_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the gamma distribution with parameters a and b.

19.15 The Flat (Uniform) Distribution

[Function] double gsl_ran_flat (const gsl rng * r, double a, double b)

This function returns a random variate from the flat (uniform) distribution from a to b. The distribution is,

p(x)dx = \frac{1}{(b − a)}dx

if a ≤ x < b and 0 otherwise.

[Function] double gsl_ran_flat_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for a uniform distribution from a to b, using the formula given above.

a = 1.2, b = 4.8 a = 0.5, b = 2.5 Flat Distribution x p(x)  5 4 3 2 1 0 1 0.5 0

[Function] double gsl_cdf_flat_P (double x, double a, double b)

[Function] double gsl_cdf_flat_Q (double x, double a, double b)

[Function] double gsl_cdf_flat_Pinv (double P, double a, double b)

[Function] double gsl_cdf_flat_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for a uniform distribution from a to b.

19.16 The Lognormal Distribution

[Function] double gsl_ran_lognormal (const gsl rng * r, double zeta, double

   sigma)

This function returns a random variate from the lognormal distribution. The distribution function is,

p(x)dx = \frac{1}{x\sqrt{2πσ^{2}}} exp(−(ln(x) − ζ)^{2}/2σ^{2})dx

for x > 0.

[Function] double gsl_ran_lognormal_pdf (double x, double zeta, double
   sigma)

This function computes the probability density p(x) at x for a lognormal distribution with parameters zeta and sigma, using the formula given above.

ζ = 1, σ = 1 ζ = 0, σ = 1 Lognormal Distribution x p(x)  3 2 1 0 0.5 0

[Function] double gsl_cdf_lognormal_P (double x, double zeta, double sigma)
[Function] double gsl_cdf_lognormal_Q (double x, double zeta, double sigma)
[Function] double gsl_cdf_lognormal_Pinv (double P, double zeta, double
   sigma)
[Function] double gsl_cdf_lognormal_Qinv (double Q, double zeta, double
   sigma)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the lognormal distribution with parameters zeta and sigma.

19.17 The Chi-squared Distribution

The chi-squared distribution arises in statistics. If Y_{i} are n independent gaussian random variates with unit variance then the sum-of-squares,

X_{i} = ∑_{i} Y_{i}^{2}

has a chi-squared distribution with n degrees of freedom.

[Function] double gsl_ran_chisq (const gsl rng * r, double nu)

This function returns a random variate from the chi-squared distribution with nu degrees of freedom. The distribution function is,

p(x)dx = \frac{1}{2Γ(ν/2)}(x/2)^{ν/2−1} exp(−x/2)dx

for x ≥ 0.

[Function] double gsl_ran_chisq_pdf (double x, double nu)

This function computes the probability density p(x) at x for a chi-squared distribution with nu degrees of freedom, using the formula given above.

ν = 3 ν = 2 ν = 1 Chi-squared Distribution x p(x)  3 2 1 0 1 0.5 0

[Function] double gsl_cdf_chisq_P (double x, double nu)

[Function] double gsl_cdf_chisq_Q (double x, double nu)

[Function] double gsl_cdf_chisq_Pinv (double P, double nu)
[Function] double gsl_cdf_chisq_Qinv (double Q, double nu)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the chi-squared distribution with nu degrees of freedom.

19.18 The F-distribution

The F-distribution arises in statistics. If Y_{1} and Y_{2} are chi-squared deviates with ν_{1} and ν_{2} degrees of freedom then the ratio,

X = \frac{(Y_{1}/ν_{1})}{(Y_{2}/ν_{2})}

has an F-distribution F (x; ν_{1}, ν_{2}).

[Function] double gsl_ran_fdist (const gsl rng * r, double nu1, double nu2)

This function returns a random variate from the F-distribution with degrees of freedom nu1 and nu2. The distribution function is,

p(x)dx = \frac{Γ((ν_{1} + ν_{2})/2)}{Γ(ν_{1}/2)Γ(ν_{2}/2)}ν_{1}^{ν_{1}/2}ν_{2}^{ν_{2}/2}x^{ν_{1}/2−1}(ν_{2} + ν_{1}x)^{−ν_{1}/2−ν_{2}/2}

for x ≥ 0.

[Function] double gsl_ran_fdist_pdf (double x, double nu1, double nu2)

This function computes the probability density p(x) at x for an F-distribution with nu1 and nu2 degrees of freedom, using the formula given above.

ν_{1} = 3, ν_{2} = 2 ν_{1} = 1, ν_{2} = 1 F-Distribution x p(x)  2 1 0 1 0.5 0

[Function] double gsl_cdf_fdist_P (double x, double nu1, double nu2)

[Function] double gsl_cdf_fdist_Q (double x, double nu1, double nu2)

[Function] double gsl_cdf_fdist_Pinv (double P, double nu1, double nu2)

[Function] double gsl_cdf_fdist_Qinv (double Q, double nu1, double nu2)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the F-distribution with nu1 and nu2 degrees of freedom.

19.19 The t-distribution

The t-distribution arises in statistics. If Y_{1} has a normal distribution and Y_{2} has a chisquared distribution with ν degrees of freedom then the ratio,

X = \frac{Y_{1}}{\sqrt{Y_{2}/ν}}

has a t-distribution t(x; ν) with ν degrees of freedom.

[Function] double gsl_ran_tdist (const gsl rng * r, double nu)

This function returns a random variate from the t-distribution. The distribution function is,

p(x)dx = \frac{Γ((ν + 1)/2)}{\sqrt{πν}Γ(ν/2)}(1 + x^{2}/ν)^{−(ν+1)/2}dx

for −∞ < x < +∞.

[Function] double gsl_ran_tdist_pdf (double x, double nu)

This function computes the probability density p(x) at x for a t-distribution with nu degrees of freedom, using the formula given above.

ν_{1} = 5 ν_{1} = 1 Student’s t distribution x p(x)  4 3 2 1 0 -1 -2 -3 -4 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_tdist_P (double x, double nu)

[Function] double gsl_cdf_tdist_Q (double x, double nu)

[Function] double gsl_cdf_tdist_Pinv (double P, double nu)
[Function] double gsl_cdf_tdist_Qinv (double Q, double nu)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the t-distribution with nu degrees of freedom.

19.20 The Beta Distribution

[Function] double gsl_ran_beta (const gsl rng * r, double a, double b)

This function returns a random variate from the beta distribution. The distribution function is,

p(x)dx = \frac{Γ(a + b)}{Γ(a)Γ(b)}x^{a−1}(1 − x)^{b−1}dx

for 0 ≤ x ≤ 1.

[Function] double gsl_ran_beta_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for a beta distribution with parameters a and b, using the formula given above.

a = 1, b = 4 a = 4, b = 1 a = 2, b = 2 Beta Distribution x p(x)  1 0.75 0.5 0.25 0 4 3 2 1 0

[Function] double gsl_cdf_beta_P (double x, double a, double b)

[Function] double gsl_cdf_beta_Q (double x, double a, double b)

[Function] double gsl_cdf_beta_Pinv (double P, double a, double b)

[Function] double gsl_cdf_beta_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the beta distribution with parameters a and b.

19.21 The Logistic Distribution

[Function] double gsl_ran_logistic (const gsl rng * r, double a)

This function returns a random variate from the logistic distribution. The distribution function is,

p(x)dx = \frac{exp(−x/a)}{a(1 + exp(−x/a))^{2}} dx

for −∞ < x < +∞.

[Function] double gsl_ran_logistic_pdf (double x, double a)

This function computes the probability density p(x) at x for a logistic distribution with scale parameter a, using the formula given above.

a = 2 a = 1 Logistic Distribution x p(x)  5 4 3 2 1 0 -1 -2 -3 -4 -5 0.3 0.2 0.1 0

[Function] double gsl_cdf_logistic_P (double x, double a)
[Function] double gsl_cdf_logistic_Q (double x, double a)
[Function] double gsl_cdf_logistic_Pinv (double P, double a)
[Function] double gsl_cdf_logistic_Qinv (double Q, double a)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the logistic distribution with scale parameter a.

19.22 The Pareto Distribution

[Function] double gsl_ran_pareto (const gsl rng * r, double a, double b)

This function returns a random variate from the Pareto distribution of order a. The distribution function is,

p(x)dx = (a/b)/(x/b)^{a+1}dx

for x ≥ b.

[Function] double gsl_ran_pareto_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for a Pareto distribution with exponent a and scale b, using the formula given above.

a = 3, b = 2 a = 1, b = 1 Pareto Distribution x p(x)  5 4 3 2 1 0 2 1.5 1 0.5 0

[Function] double gsl_cdf_pareto_P (double x, double a, double b)

[Function] double gsl_cdf_pareto_Q (double x, double a, double b)

[Function] double gsl_cdf_pareto_Pinv (double P, double a, double b)

[Function] double gsl_cdf_pareto_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Pareto distribution with exponent a and scale b.

19.23 Spherical Vector Distributions

The spherical distributions generate random vectors, located on a spherical surface. They can be used as random directions, for example in the steps of a random walk.

[Function] void gsl_ran_dir_2d (const gsl rng * r, double * x, double * y)

[Function] void gsl_ran_dir_2d_trig_method (const gsl rng * r, double * x,

double * y) This function returns a random direction vector v = (x,y) in two dimensions. The vector is normalized such that |v|^{2} = x^{2} + y^{2} = 1. The obvious way to do this is to take a uniform random number between 0 and 2π and let x and y be the sine and cosine respectively. Two trig functions would have been expensive in the old days, but with modern hardware implementations, this is sometimes the fastest way to go. This is the case for the Pentium (but not the case for the Sun Sparcstation). One can avoid the trig evaluations by choosing x and y in the interior of a unit circle (choose them at random from the interior of the enclosing square, and then reject those that are outside the unit circle), and then dividing by \sqrt{x^{2} + y^{2}}. A much cleverer approach, attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise 23), requires neither trig nor a square root. In this approach, u and v are chosen at random from the interior of a unit circle, and then x = (u^{2} − v^{2})/(u^{2} + v^{2}) and y = 2uv/(u^{2} + v^{2}).

[Function] void gsl_ran_dir_3d (const gsl rng * r, double * x, double * y, double

* z) This function returns a random direction vector v = (x,y,z) in three dimensions. The vector is normalized such that |v|^{2} = x^{2} +y^{2} +z^{2} = 1. The method employed is due to Robert E. Knop (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136. It uses the surprising fact that the distribution projected along any axis is actually uniform (this is only true for 3 dimensions).

[Function] void gsl_ran_dir_nd (const gsl rng * r, size t n, double * x)

This function returns a random direction vector v = (x_{1}, x_{2}, …, x_{n}) in n dimensions. The vector is normalized such that |v|^{2} = x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n}^{2} = 1. The method uses the fact that a multivariate gaussian distribution is spherically symmetric. Each component is generated to have a gaussian distribution, and then the components are normalized. The method is described by Knuth, v2, 3rd ed, p135–136, and attributed to G. W. Brown, Modern Mathematics for the Engineer (1956).

19.24 The Weibull Distribution

[Function] double gsl_ran_weibull (const gsl rng * r, double a, double b)

This function returns a random variate from the Weibull distribution. The distribution function is,

p(x)dx = \frac{b}{a^{b}} x^{b−1} exp(−(x/a)^{b})dx

for x ≥ 0.

[Function] double gsl_ran_weibull_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for a Weibull distribution with scale a and exponent b, using the formula given above.

a = 2, b = 3 a = 1, b = 2 a = 1, b = 1 Weibull Distribution x p(x)  2 1.5 1 0.5 0 1.5 1 0.5 0

[Function] double gsl_cdf_weibull_P (double x, double a, double b)

[Function] double gsl_cdf_weibull_Q (double x, double a, double b)

[Function] double gsl_cdf_weibull_Pinv (double P, double a, double b)

[Function] double gsl_cdf_weibull_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Weibull distribution with scale a and exponent b.

19.25 The Type-1 Gumbel Distribution

[Function] double gsl_ran_gumbel1 (const gsl rng * r, double a, double b)

This function returns a random variate from the Type-1 Gumbel distribution. The Type-1 Gumbel distribution function is,

p(x)dx = ab exp(−(b exp(−ax) + ax))dx

for −∞ < x < ∞.

[Function] double gsl_ran_gumbel1_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for a Type-1 Gumbel distribution with parameters a and b, using the formula given above.

Type 1, a = 1, b = 1 Type 1 Gumbel Distribution x p(x)  5 4 3 2 1 0 -1 -2 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_gumbel1_P (double x, double a, double b)

[Function] double gsl_cdf_gumbel1_Q (double x, double a, double b)

[Function] double gsl_cdf_gumbel1_Pinv (double P, double a, double b)

[Function] double gsl_cdf_gumbel1_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Type-1 Gumbel distribution with parameters a and b.

19.26 The Type-2 Gumbel Distribution

[Function] double gsl_ran_gumbel2 (const gsl rng * r, double a, double b)

This function returns a random variate from the Type-2 Gumbel distribution. The Type-2 Gumbel distribution function is,

p(x)dx = abx^{−a−1} exp(−bx^{−a})dx

for 0 < x < ∞.

[Function] double gsl_ran_gumbel2_pdf (double x, double a, double b)

This function computes the probability density p(x) at x for a Type-2 Gumbel distribution with parameters a and b, using the formula given above.

Type 2, a = 1, b = 1 Type 2 Gumbel Distribution x p(x)  2 1.5 1 0.5 0 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_gumbel2_P (double x, double a, double b)

[Function] double gsl_cdf_gumbel2_Q (double x, double a, double b)

[Function] double gsl_cdf_gumbel2_Pinv (double P, double a, double b)

[Function] double gsl_cdf_gumbel2_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Type-2 Gumbel distribution with parameters a and b.

19.27 The Dirichlet Distribution

[Function] void gsl_ran_dirichlet (const gsl rng * r, size t K, const double

alpha[], double theta[])

This function returns an array of K random variates from a Dirichlet distribution of order K-1. The distribution function is

p(θ_{1}, …, θ_{K}) dθ_{1} ⋯ dθ_{K} = \frac{1}{Z} ∏_{i=1}^{K} θ_{i}^{α_{i}−1} δ(1 − ∑_{i=1}^{K} θ_{i})dθ_{1} ⋯ dθ_{K}

for θ_{i} ≥ 0 and α_{i} ≥ 0. The delta function ensures that ∑ θ_{i} = 1. The normalization factor Z is

Z = \frac{∏_{i=1}^{K} Γ(α_{i})}{Γ(∑_{i=1}^{K} α_{i})}

The random variates are generated by sampling K values from gamma distributions with parameters a = α_{i}, b = 1, and renormalizing. See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).

[Function] double gsl_ran_dirichlet_pdf (size t K, const double alpha[], const

double theta[])

This function computes the probability density p(θ_{1}, …, θ_{K}) at theta[K] for a Dirichlet distribution with parameters alpha[K], using the formula given above.

[Function] double gsl_ran_dirichlet_lnpdf (size t K, const double alpha[],

const double theta[]) This function computes the logarithm of the probability density p(θ_{1}, …, θ_{K}) for a Dirichlet distribution with parameters alpha[K].

19.28 General Discrete Distributions

Given K discrete events with different probabilities P [k], produce a random value k consistent with its probability.

The obvious way to do this is to preprocess the probability list by generating a cumulative probability array with K + 1 elements:

\begin{aligned}C[0] & = 0 \\ C[k + 1] & = C[k] + P [k]. \end{aligned}

Note that this construction produces C[K] = 1. Now choose a uniform deviate u between 0 and 1, and find the value of k such that C[k] ≤ u < C[k + 1]. Although this in principle requires of order log K steps per random number generation, they are fast steps, and if you use something like ⌊uK⌋ as a starting point, you can often do pretty well.

But faster methods have been devised. Again, the idea is to preprocess the probability list, and save the result in some form of lookup table; then the individual calls for a random discrete event can go rapidly. An approach invented by G. Marsaglia (Generating discrete random numbers in a computer, Comm ACM 6, 37–38 (1963)) is very clever, and readers interested in examples of good algorithm design are directed to this short and well-written paper. Unfortunately, for large K, Marsaglia’s lookup table can be quite large.

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

This method can be used to speed up some of the discrete random number generators below, such as the binomial distribution. To use it for something like the Poisson Distribution, a modification would have to be made, since it only takes a finite set of K outcomes.

[Function] gsl_ran_discrete_t * gsl_ran_discrete_preproc (size t K, const

double * P) This function returns a pointer to a structure that contains the lookup table for the discrete random number generator. The array P[] contains the probabilities of the discrete events; these array elements must all be positive, but they needn’t add up to one (so you can think of them more generally as “weights”)—the preprocessor will normalize appropriately. This return value is used as an argument for the gsl_ran_ discrete function below.

[Function] size_t gsl_ran_discrete (const gsl rng * r, const gsl ran discrete t *

        g)

After the preprocessor, above, has been called, you use this function to get the discrete random numbers.

[Function] double gsl_ran_discrete_pdf (size t k, const gsl ran discrete t * g)

Returns the probability P [k] of observing the variable k. Since P [k] is not stored as part of the lookup table, it must be recomputed; this computation takes O(K), so if K is large and you care about the original array P [k] used to create the lookup table, then you should just keep this original array P [k] around.

[Function] void gsl_ran_discrete_free (gsl ran discrete t * g)

De-allocates the lookup table pointed to by g.

19.29 The Poisson Distribution

[Function] unsigned int gsl_ran_poisson (const gsl rng * r, double mu)

This function returns a random integer from the Poisson distribution with mean mu. The probability distribution for Poisson variates is,

p(k) = \frac{μ^{k}}{k!} exp(−μ)

for k ≥ 0.

[Function] double gsl_ran_poisson_pdf (unsigned int k, double mu)

This function computes the probability p(k) of obtaining k from a Poisson distribution with mean mu, using the formula given above.

μ = 2.5 Poisson Distribution k p(k)  10 9 8 7 6 5 4 3 2 1 0 0.3 0.2 0.1 0

[Function] double gsl_cdf_poisson_P (unsigned int k, double mu)

[Function] double gsl_cdf_poisson_Q (unsigned int k, double mu)

These functions compute the cumulative distribution functions P (k), Q(k) for the Poisson distribution with parameter mu.

19.30 The Bernoulli Distribution

[Function] unsigned int gsl_ran_bernoulli (const gsl rng * r, double p)

This function returns either 0 or 1, the result of a Bernoulli trial with probability p. The probability distribution for a Bernoulli trial is,

\begin{aligned}p(0) & = 1 − p \\ p(1) & = p \end{aligned}

[Function] double gsl_ran_bernoulli_pdf (unsigned int k, double p)

This function computes the probability p(k) of obtaining k from a Bernoulli distribution with probability parameter p, using the formula given above.

p = 0.7 Bernoulli Trial k p(k)  1 0 1 0.5 0

19.31 The Binomial Distribution

[Function] unsigned int gsl_ran_binomial (const gsl rng * r, double p,

unsigned int n) This function returns a random integer from the binomial distribution, the number of successes in n independent trials with probability p. The probability distribution for binomial variates is,

p(k) = \frac{n!}{k!(n − k)!}p^{k}(1 − p)^{n−k}

for 0 ≤ k ≤ n.

[Function] double gsl_ran_binomial_pdf (unsigned int k, double p, unsigned int

   n)

This function computes the probability p(k) of obtaining k from a binomial distribution with parameters p and n, using the formula given above.

p = 0.5, n = 9 Binomial Distribution k p(k)  10 9 8 7 6 5 4 3 2 1 0 0.3 0.2 0.1 0

[Function] double gsl_cdf_binomial_P (unsigned int k, double p, unsigned int n)

[Function] double gsl_cdf_binomial_Q (unsigned int k, double p, unsigned int n)

These functions compute the cumulative distribution functions P (k), Q(k) for the binomial distribution with parameters p and n.

19.32 The Multinomial Distribution

[Function] void gsl_ran_multinomial (const gsl rng * r, size t K, unsigned int N,

const double p[], unsigned int n[]) This function returns an array of K random variates from a multinomial distribution. The distribution function is,

P (n_{1}, n_{2}, ⋯, n_{K}) = \frac{N!}{n_{1}!n_{2}! ⋯ n_{K}!} p_{1}^{n_{1}}p_{2}^{n_{2}} ⋯ p_{K}^{n_{K}}

where (n_{1}, n_{2}, …, n_{K}) are nonnegative integers with ∑_{k=1}^{K} n_{k} = N, and (p_{1}, p_{2}, …, p_{K}) is a probability distribution with ∑ p_{i} = 1. If the array p[K] is not normalized then its entries will be treated as weights and normalized appropriately. Random variates are generated using the conditional binomial method (see C.S. David, The computer generation of multinomial random variates, Comp. Stat. Data Anal. 16 (1993) 205–217 for details).

[Function] double gsl_ran_multinomial_pdf (size t K, const double p[], const

unsigned int n[]) This function computes the probability P (n_{1}, n_{2}, …, n_{K}) of sampling n[K] from a multinomial distribution with parameters p[K], using the formula given above.

[Function] double gsl_ran_multinomial_lnpdf (size t K, const double p[], const

unsigned int n[]) This function returns the logarithm of the probability for the multinomial distribution P (n_{1}, n_{2}, …, n_{K}) with parameters p[K].

19.33 The Negative Binomial Distribution

[Function] unsigned int gsl_ran_negative_binomial (const gsl rng * r,

double p, double n) This function returns a random integer from the negative binomial distribution, the number of failures occurring before n successes in independent trials with probability p of success. The probability distribution for negative binomial variates is,

p(k) = \frac{Γ(n + k)}{Γ(k + 1)Γ(n)}p^{n}(1 − p)^{k}

Note that n is not required to be an integer.

[Function] double gsl_ran_negative_binomial_pdf (unsigned int k, double p,

double n) This function computes the probability p(k) of obtaining k from a negative binomial distribution with parameters p and n, using the formula given above.

p = 0.5, n = 3.5 Negative Binomial Distribution k p(k)  10 9 8 7 6 5 4 3 2 1 0 0.3 0.2 0.1 0

[Function] double gsl_cdf_negative_binomial_P (unsigned int k, double p,

double n)

[Function] double gsl_cdf_negative_binomial_Q (unsigned int k, double p,

double n) These functions compute the cumulative distribution functions P (k), Q(k) for the negative binomial distribution with parameters p and n.

19.34 The Pascal Distribution

[Function] unsigned int gsl_ran_pascal (const gsl rng * r, double p, unsigned

int n) This function returns a random integer from the Pascal distribution. The Pascal distribution is simply a negative binomial distribution with an integer value of n.

p(k) = \frac{(n + k − 1)!}{k!(n − 1)!}p^{n}(1 − p)^{k}

for k ≥ 0

[Function] double gsl_ran_pascal_pdf (unsigned int k, double p, unsigned int n)

This function computes the probability p(k) of obtaining k from a Pascal distribution with parameters p and n, using the formula given above.

p = 0.5, n = 3 Pascal Distribution k p(k)  10 9 8 7 6 5 4 3 2 1 0 0.3 0.2 0.1 0

[Function] double gsl_cdf_pascal_P (unsigned int k, double p, unsigned int n)

[Function] double gsl_cdf_pascal_Q (unsigned int k, double p, unsigned int n)

These functions compute the cumulative distribution functions P (k), Q(k) for the Pascal distribution with parameters p and n.

19.35 The Geometric Distribution

[Function] unsigned int gsl_ran_geometric (const gsl rng * r, double p)

This function returns a random integer from the geometric distribution, the number of independent trials with probability p until the first success. The probability distribution for geometric variates is,

p(k) = p(1 − p)^{k−1}

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

[Function] double gsl_ran_geometric_pdf (unsigned int k, double p)

This function computes the probability p(k) of obtaining k from a geometric distribution with probability parameter p, using the formula given above.

p = 0.5 Geometric Distribution k p(k)  5 4 3 2 1 0 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_geometric_P (unsigned int k, double p)

[Function] double gsl_cdf_geometric_Q (unsigned int k, double p)

These functions compute the cumulative distribution functions P (k), Q(k) for the geometric distribution with parameter p.

19.36 The Hypergeometric Distribution

[Function] unsigned int gsl_ran_hypergeometric (const gsl rng * r, unsigned

int n1, unsigned int n2, unsigned int t) This function returns a random integer from the hypergeometric distribution. The probability distribution for hypergeometric random variates is,

p(k) = C(n_{1}, k)C(n_{2}, t − k)/C(n_{1} + n_{2}, t)

where C(a, b) = a!/(b!(a − b)!) and t ≤ n_{1} + n_{2}. The domain of k is max(0, t − n_{2}), …, min(t, n_{1}). If a population contains n_{1} elements of “type 1” and n_{2} elements of “type 2” then the hypergeometric distribution gives the probability of obtaining k elements of “type 1” in t samples from the population without replacement.

[Function] double gsl_ran_hypergeometric_pdf (unsigned int k, unsigned int

n1, unsigned int n2, unsigned int t) This function computes the probability p(k) of obtaining k from a hypergeometric distribution with parameters n1, n2, t, using the formula given above.

n1 = 5, n2 = 20, t = 3 Hypergeometric Distribution k p(k)  10 9 8 7 6 5 4 3 2 1 0 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

[Function] double gsl_cdf_hypergeometric_P (unsigned int k, unsigned int n1,

unsigned int n2, unsigned int t)

[Function] double gsl_cdf_hypergeometric_Q (unsigned int k, unsigned int n1,

unsigned int n2, unsigned int t) These functions compute the cumulative distribution functions P (k), Q(k) for the hypergeometric distribution with parameters n1, n2 and t.

19.37 The Logarithmic Distribution

[Function] unsigned int gsl_ran_logarithmic (const gsl rng * r, double p)

This function returns a random integer from the logarithmic distribution. The probability distribution for logarithmic random variates is,

p(k) = \frac{−1}{log(1 − p)}(\frac{p^{k}}{k} )

for k ≥ 1.

[Function] double gsl_ran_logarithmic_pdf (unsigned int k, double p)

This function computes the probability p(k) of obtaining k from a logarithmic distribution with probability parameter p, using the formula given above.

p = 0.7 Logarithmic Distribution k p(k)  10 9 8 7 6 5 4 3 2 1 0 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

19.38 Shuffling and Sampling

The following functions allow the shuffling and sampling of a set of objects. The algorithms rely on a random number generator as a source of randomness and a poor quality generator can lead to correlations in the output. In particular it is important to avoid generators with a short period. For more information see Knuth, v2, 3rd ed, Section 3.4.2, “Random Sampling and Shuffling”.

[Function] void gsl_ran_shuffle (const gsl rng * r, void * base, size t n, size t

   size)

This function randomly shuffles the order of n objects, each of size size, stored in the array base[0..n-1]. The output of the random number generator r is used to produce the permutation. The algorithm generates all possible n! permutations with equal probability, assuming a perfect source of random numbers.

The following code shows how to shuffle the numbers from 0 to 51,

     int a[52];
     for (i = 0; i < 52; i++)
       {
         a[i] = i;
       }
     gsl_ran_shuffle (r, a, 52, sizeof (int));

[Function] int gsl_ran_choose (const gsl rng * r, void * dest, size t k, void *

src, size t n, size t size) This function fills the array dest[k] with k objects taken randomly from the n elements of the array src[0..n-1]. The objects are each of size size. The output of the random number generator r is used to make the selection. The algorithm ensures all possible samples are equally likely, assuming a perfect source of randomness.

The objects are sampled without replacement, thus each object can only appear once in dest[k]. It is required that k be less than or equal to n. The objects in dest will be in the same relative order as those in src. You will need to call gsl_ran_shuffle(r, dest, n, size) if you want to randomize the order.

The following code shows how to select a random sample of three unique numbers from the set 0 to 99,

     double a[3], b[100];
     for (i = 0; i < 100; i++)
       {
         b[i] = (double) i;
       }
     gsl_ran_choose (r, a, 3, b, 100, sizeof (double));

[Function] void gsl_ran_sample (const gsl rng * r, void * dest, size t k, void *

src, size t n, size t size) This function is like gsl_ran_choose but samples k items from the original array of n items src with replacement, so the same object can appear more than once in the output sequence dest. There is no requirement that k be less than n in this case.

19.39 Examples

The following program demonstrates the use of a random number generator to produce variates from a distribution. It prints 10 samples from the Poisson distribution with a mean of 3.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
double mu = 3.0;
/* create a generator chosen by the
   environment variable GSL_RNG_TYPE */
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
/* print n random variates chosen from
   the poisson distribution with mean
   parameter mu */
for (i = 0; i < n; i++)
  {
    unsigned int k = gsl_ran_poisson (r, mu);
    printf (" %u", k);
  }
printf ("\n");
return 0;
}

If the library and header files are installed under ‘/usr/local’ (the default location) then the program can be compiled with these options,

$ gcc -Wall demo.c -lgsl -lgslcblas -lm

Here is the output of the program,

$ ./a.out
 2 5 5 2 1 0 3 4 1 1

The variates depend on the seed used by the generator. The seed for the default generator type gsl_rng_default can be changed with the GSL_RNG_SEED environment variable to produce a different stream of variates,

$ GSL_RNG_SEED=123 ./a.out
GSL_RNG_SEED=123
 4 5 6 3 3 1 4 2 5 5

The following program generates a random walk in two dimensions.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
int
main (void)
{
  int i;
  double x = 0, y = 0, dx, dy;
  const gsl_rng_type * T;
  gsl_rng * r;
  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);
  printf ("%g %g\n", x, y);
  for (i = 0; i < 10; i++)
    {
      gsl_ran_dir_2d (r, &dx, &dy);
      x += dx; y += dy;
      printf ("%g %g\n", x, y);
    }
  return 0;
}

Here is the output from the program, three 10-step random walks from the origin,

Random walk k p(k)  5 4 3 2 1 0 -1 -2 -3 -4 -5 5 4 3 2 1 0 -1 -2 -3 -4 -5

The following program computes the upper and lower cumulative distribution functions for the standard normal distribution at x = 2.

#include <stdio.h>
#include <gsl/gsl_cdf.h>
int
main (void)
{
double P, Q;
double x = 2.0;
P = gsl_cdf_ugaussian_P (x);
printf ("prob(x < %f) = %f\n", x, P);
Q = gsl_cdf_ugaussian_Q (x);
printf ("prob(x > %f) = %f\n", x, Q);
x = gsl_cdf_ugaussian_Pinv (P);
printf ("Pinv(%f) = %f\n", P, x);
x = gsl_cdf_ugaussian_Qinv (Q);
printf ("Qinv(%f) = %f\n", Q, x);
return 0;
}

Here is the output of the program,

prob(x < 2.000000) = 0.977250
prob(x > 2.000000) = 0.022750
Pinv(0.977250) = 2.000000

     Qinv(0.022750) = 2.000000

19.40 References and Further Reading

For an encyclopaedic coverage of the subject readers are advised to consult the book Non-Uniform Random Variate Generation by Luc Devroye. It covers every imaginable distribution and provides hundreds of algorithms.

Luc Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, ISBN 0-387-96305-7.

The subject of random variate generation is also reviewed by Knuth, who describes algorithms for all the major distributions.

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

The Particle Data Group provides a short review of techniques for generating distributions of random numbers in the “Monte Carlo” section of its Annual Review of Particle Physics.

Review of Particle Properties R.M. Barnett et al., Physical Review D54, 1 (1996)

    http://pdg.lbl.gov/.

The Review of Particle Physics is available online in postscript and pdf format. An overview of methods used to compute cumulative distribution functions can be found in Statistical Computing by W.J. Kennedy and J.E. Gentle. Another general reference is Elements of Statistical Computing by R.A. Thisted.

William E. Kennedy and James E. Gentle, Statistical Computing (1980), Marcel Dekker, ISBN 0-8247-6898-1. Ronald A. Thisted, Elements of Statistical Computing (1988), Chapman & Hall, ISBN 0-412-01371-1.

The cumulative distribution functions for the Gaussian distribution are based on the following papers,

Rational Chebyshev Approximations Using Linear Equations, W.J. Cody, W. Fraser, J.F. Hart. Numerische Mathematik 12, 242–251 (1968). Rational Chebyshev Approximations for the Error Function, W.J. Cody. Mathematics of Computation 23, n107, 631–637 (July 1969).

20 Statistics

This chapter describes the statistical functions in the library. The basic statistical functions include routines to compute the mean, variance and standard deviation. More advanced functions allow you to calculate absolute deviations, skewness, and kurtosis as well as the median and arbitrary percentiles. The algorithms use recurrence relations to compute average quantities in a stable way, without large intermediate values that might overflow.

The functions are available in versions for datasets in the standard floating-point and integer types. The versions for double precision floating-point data have the prefix gsl_stats and are declared in the header file ‘gsl_statistics_double.h’. The versions for integer data have the prefix gsl_stats_int and are declared in the header file

‘gsl_statistics_int.h’.

20.1 Mean, Standard Deviation and Variance

[Function] double gsl_stats_mean (const double data[], size t stride, size t n)

This function returns the arithmetic mean of data, a dataset of length n with stride stride. The arithmetic mean, or sample mean, is denoted by μ̂ and defined as,

μ̂ = \frac{1}{N} ∑ x_{i}

where x_{i} are the elements of the dataset data. For samples drawn from a gaussian distribution the variance of μ̂ is σ^{2}/N.

[Function] double gsl_stats_variance (const double data[], size t stride,

size t n) This function returns the estimated, or sample, variance of data, a dataset of length n with stride stride. The estimated variance is denoted by σ̂^{2} and is defined by,

σ̂^{2} = \frac{1}{(N − 1)} ∑(x_{i} − μ̂)^{2}

where x_{i} are the elements of the dataset data. Note that the normalization factor of 1/(N −1) results from the derivation of σ̂^{2} as an unbiased estimator of the population variance σ^{2}. For samples drawn from a gaussian distribution the variance of σ̂^{2} itself is 2σ^{4}/N.

This function computes the mean via a call to gsl_stats_mean. If you have already computed the mean then you can pass it directly to gsl_stats_variance_m.

[Function] double gsl_stats_variance_m (const double data[], size t stride,

size t n, double mean) This function returns the sample variance of data relative to the given value of mean. The function is computed with μ̂ replaced by the value of mean that you supply,

σ̂^{2} = \frac{1}{(N − 1)} ∑(x_{i} − mean)^{2}

[Function] double gsl_stats_sd (const double data[], size t stride, size t n)

[Function] double gsl_stats_sd_m (const double data[], size t stride, size t n,

double mean) The standard deviation is defined as the square root of the variance. These functions return the square root of the corresponding variance functions above.

[Function] double gsl_stats_variance_with_fixed_mean (const double

data[], size t stride, size t n, double mean) This function computes an unbiased estimate of the variance of data when the population mean mean of the underlying distribution is known a priori. In this case the estimator for the variance uses the factor 1/N and the sample mean μ̂ is replaced by the known population mean μ,

σ̂^{2} = \frac{1}{N} ∑(x_{i} − μ)^{2}

[Function] double gsl_stats_sd_with_fixed_mean (const double data[], size t

stride, size t n, double mean) This function calculates the standard deviation of data for a fixed population mean mean. The result is the square root of the corresponding variance function.

20.2 Absolute deviation

[Function] double gsl_stats_absdev (const double data[], size t stride, size t

   n)

This function computes the absolute deviation from the mean of data, a dataset of length n with stride stride. The absolute deviation from the mean is defined as,

absdev = \frac{1}{N} ∑ |x_{i} − μ̂|

where x_{i} are the elements of the dataset data. The absolute deviation from the mean provides a more robust measure of the width of a distribution than the variance. This function computes the mean of data via a call to gsl_stats_mean.

[Function] double gsl_stats_absdev_m (const double data[], size t stride,

size t n, double mean) This function computes the absolute deviation of the dataset data relative to the given value of mean,

absdev = \frac{1}{N} ∑ |x_{i} − mean|

This function is useful if you have already computed the mean of data (and want to avoid recomputing it), or wish to calculate the absolute deviation relative to another value (such as zero, or the median).

20.3 Higher moments (skewness and kurtosis)

[Function] double gsl_stats_skew (const double data[], size t stride, size t n)

This function computes the skewness of data, a dataset of length n with stride stride. The skewness is defined as,

skew = \frac{1}{N} ∑ (\frac{x_{i} − μ̂}{σ̂})^{3}

where x_{i} are the elements of the dataset data. The skewness measures the asymmetry of the tails of a distribution.

The function computes the mean and estimated standard deviation of data via calls

to gsl_stats_mean and gsl_stats_sd.
[Function] double gsl_stats_skew_m_sd (const double data[], size t stride,

size t n, double mean, double sd) This function computes the skewness of the dataset data using the given values of the mean mean and standard deviation sd,

skew = \frac{1}{N} ∑ (\frac{x_{i} − mean}{sd})^{3}

These functions are useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.

[Function] double gsl_stats_kurtosis (const double data[], size t stride,

size t n) This function computes the kurtosis of data, a dataset of length n with stride stride. The kurtosis is defined as,

kurtosis = (\frac{1}{N} ∑ (\frac{x_{i} − μ̂}{σ̂})^{4}) − 3

The kurtosis measures how sharply peaked a distribution is, relative to its width. The kurtosis is normalized to zero for a gaussian distribution.

[Function] double gsl_stats_kurtosis_m_sd (const double data[], size t

stride, size t n, double mean, double sd) This function computes the kurtosis of the dataset data using the given values of the mean mean and standard deviation sd,

kurtosis = \frac{1}{N} (∑ (\frac{x_{i} − mean}{sd})^{4}) − 3

This function is useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.

20.4 Autocorrelation

[Function] double gsl_stats_lag1_autocorrelation (const double data[],

const size t stride, const size t n) This function computes the lag-1 autocorrelation of the dataset data.

a_{1} = \frac{∑_{i=1}^{n}(x_{i} − μ̂)(x_{i−1} − μ̂)}{∑_{i=1}^{n}(x_{i} − μ̂)(x_{i} − μ̂)}

[Function] double gsl_stats_lag1_autocorrelation_m (const double data[],

const size t stride, const size t n, const double mean) This function computes the lag-1 autocorrelation of the dataset data using the given value of the mean mean.

20.5 Covariance

[Function] double gsl_stats_covariance (const double data1[], const size t

stride1, const double data2[], const size t stride2, const size t n) This function computes the covariance of the datasets data1 and data2 which must both be of the same length n.

covar = \frac{1}{(n − 1)} ∑_{i=1}^{n}(x_{i} − x̂)(y_{i} − ŷ)

[Function] double gsl_stats_covariance_m (const double data1[], const size t

stride1, const double data2[], const size t n, const double mean1, const double mean2)

This function computes the covariance of the datasets data1 and data2 using the given values of the means, mean1 and mean2. This is useful if you have already computed the means of data1 and data2 and want to avoid recomputing them.

20.6 Weighted Samples

The functions described in this section allow the computation of statistics for weighted samples. The functions accept an array of samples, x_{i}, with associated weights, w_{i}. Each sample x_{i} is considered as having been drawn from a Gaussian distribution with variance σ_{i}^{2}. The sample weight w_{i} is defined as the reciprocal of this variance, w_{i} = 1/σ_{i}^{2}. Setting a weight to zero corresponds to removing a sample from a dataset.

[Function] double gsl_stats_wmean (const double w[], size t wstride, const

double data[], size t stride, size t n) This function returns the weighted mean of the dataset data with stride stride and length n, using the set of weights w with stride wstride and length n. The weighted mean is defined as,

μ̂ = \frac{∑ w_{i}x_{i}}{∑ w_{i}}

[Function] double gsl_stats_wvariance (const double w[], size t wstride, const

double data[], size t stride, size t n) This function returns the estimated variance of the dataset data with stride stride and length n, using the set of weights w with stride wstride and length n. The estimated variance of a weighted dataset is defined as,

σ̂^{2} = \frac{∑ w_{i}}{(∑ w_{i})^{2} − ∑(w_{i}^{2})} ∑ w_{i}(x_{i} − μ̂)^{2}

Note that this expression reduces to an unweighted variance with the familiar 1/(N − 1) factor when there are N equal non-zero weights.

[Function] double gsl_stats_wvariance_m (const double w[], size t wstride,

const double data[], size t stride, size t n, double wmean) This function returns the estimated variance of the weighted dataset data using the given weighted mean wmean.

[Function] double gsl_stats_wsd (const double w[], size t wstride, const double

data[], size t stride, size t n)

The standard deviation is defined as the square root of the variance. This function returns the square root of the corresponding variance function gsl_stats_wvariance above.

[Function] double gsl_stats_wsd_m (const double w[], size t wstride, const

double data[], size t stride, size t n, double wmean) This function returns the square root of the corresponding variance function gsl_

stats_wvariance_m above.
[Function] double gsl_stats_wvariance_with_fixed_mean (const double w[],

size t wstride, const double data[], size t stride, size t n, const double

   mean)

This function computes an unbiased estimate of the variance of weighted dataset data when the population mean mean of the underlying distribution is known a priori. In this case the estimator for the variance replaces the sample mean μ̂ by the known population mean μ,

σ̂^{2} = \frac{∑ w_{i}(x_{i} − μ)^{2}}{∑ w_{i}}

[Function] double gsl_stats_wsd_with_fixed_mean (const double w[], size t

wstride, const double data[], size t stride, size t n, const double mean) The standard deviation is defined as the square root of the variance. This function returns the square root of the corresponding variance function above.

[Function] double gsl_stats_wabsdev (const double w[], size t wstride, const

double data[], size t stride, size t n) This function computes the weighted absolute deviation from the weighted mean of data. The absolute deviation from the mean is defined as,

absdev = \frac{∑ w_{i}|x_{i} − μ̂|}{∑ w_{i}}

[Function] double gsl_stats_wabsdev_m (const double w[], size t wstride, const

double data[], size t stride, size t n, double wmean) This function computes the absolute deviation of the weighted dataset data about the given weighted mean wmean.

[Function] double gsl_stats_wskew (const double w[], size t wstride, const

double data[], size t stride, size t n) This function computes the weighted skewness of the dataset data.

skew = \frac{∑ w_{i}((x_{i} − xbar)/σ)^{3}}{∑ w_{i}}

[Function] double gsl_stats_wskew_m_sd (const double w[], size t wstride,

const double data[], size t stride, size t n, double wmean, double wsd) This function computes the weighted skewness of the dataset data using the given values of the weighted mean and weighted standard deviation, wmean and wsd.

[Function] double gsl_stats_wkurtosis (const double w[], size t wstride, const

double data[], size t stride, size t n) This function computes the weighted kurtosis of the dataset data.

kurtosis = \frac{∑ w_{i}((x_{i} − xbar)/sigma)^{4}}{∑ w_{i}} − 3

[Function] double gsl_stats_wkurtosis_m_sd (const double w[], size t wstride,

const double data[], size t stride, size t n, double wmean, double wsd) This function computes the weighted kurtosis of the dataset data using the given values of the weighted mean and weighted standard deviation, wmean and wsd.

20.7 Maximum and Minimum values

The following functions find the maximum and minimum values of a dataset (or their indices). If the data contains NaNs then a NaN will be returned, since the maximum or minimum value is undefined. For functions which return an index, the location of the first NaN in the array is returned.

[Function] double gsl_stats_max (const double data[], size t stride, size t n)

This function returns the maximum value in data, a dataset of length n with stride stride. The maximum value is defined as the value of the element x_{i} which satisfies x_{i} ≥ x_{j} for all j. If you want instead to find the element with the largest absolute magnitude you will need to apply fabs or abs to your data before calling this function.

[Function] double gsl_stats_min (const double data[], size t stride, size t n)

This function returns the minimum value in data, a dataset of length n with stride stride. The minimum value is defined as the value of the element x_{i} which satisfies x_{i} ≤ x_{j} for all j. If you want instead to find the element with the smallest absolute magnitude you will need to apply fabs or abs to your data before calling this function.

[Function] void gsl_stats_minmax (double * min, double * max, const double

data[], size t stride, size t n)

This function finds both the minimum and maximum values min, max in data in a single pass.

[Function] size_t gsl_stats_max_index (const double data[], size t stride,

size t n) This function returns the index of the maximum value in data, a dataset of length n with stride stride. The maximum value is defined as the value of the element x_{i} which satisfies x_{i} ≥ x_{j} for all j. When there are several equal maximum elements then the first one is chosen.

[Function] size_t gsl_stats_min_index (const double data[], size t stride,

size t n) This function returns the index of the minimum value in data, a dataset of length n with stride stride. The minimum value is defined as the value of the element x_{i} which satisfies x_{i} ≥ x_{j} for all j. When there are several equal minimum elements then the first one is chosen.

[Function] void gsl_stats_minmax_index (size t * min_index, size t *

max_index, const double data[], size t stride, size t n) This function returns the indexes min index, max index of the minimum and maximum values in data in a single pass.

20.8 Median and Percentiles

The median and percentile functions described in this section operate on sorted data. For convenience we use quantiles, measured on a scale of 0 to 1, instead of percentiles (which use a scale of 0 to 100).

[Function] double gsl_stats_median_from_sorted_data (const double
   sorted_data[], size t stride, size t n)

This function returns the median value of sorted data, a dataset of length n with stride stride. The elements of the array must be in ascending numerical order. There are no checks to see whether the data are sorted, so the function gsl_sort should always be used first. When the dataset has an odd number of elements the median is the value of element (n − 1)/2. When the dataset has an even number of elements the median is the mean of the two nearest middle values, elements (n − 1)/2 and n/2. Since the algorithm for computing the median involves interpolation this function always returns a floatingpoint number, even for integer data types.

[Function] double gsl_stats_quantile_from_sorted_data (const double

sorted_data[], size t stride, size t n, double f)

This function returns a quantile value of sorted data, a double-precision array of length n with stride stride. The elements of the array must be in ascending numerical order. The quantile is determined by the f, a fraction between 0 and 1. For example, to compute the value of the 75th percentile f should have the value 0.75.

There are no checks to see whether the data are sorted, so the function gsl_sort should always be used first.

The quantile is found by interpolation, using the formula

quantile = (1 − δ)x_{i} + δx_{i+1}

where i is floor((n − 1)f) and δ is (n − 1)f − i.

Thus the minimum value of the array (data[0*stride]) is given by f equal to zero, the maximum value (data[(n-1)*stride]) is given by f equal to one and the median value is given by f equal to 0.5. Since the algorithm for computing quantiles involves interpolation this function always returns a floating-point number, even for integer data types.

20.9 Examples

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

#include <stdio.h>
#include <gsl/gsl_statistics.h>
int
main(void)
{
  double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
  double mean, variance, largest, smallest;
  mean     = gsl_stats_mean(data, 1, 5);
  variance = gsl_stats_variance(data, 1, 5);
  largest  = gsl_stats_max(data, 1, 5);
  smallest = gsl_stats_min(data, 1, 5);
  printf ("The dataset is %g, %g, %g, %g, %g\n",
         data[0], data[1], data[2], data[3], data[4]);
  printf ("The sample mean is %g\n", mean);
  printf ("The estimated variance is %g\n", variance);
  printf ("The largest value is %g\n", largest);
  printf ("The smallest value is %g\n", smallest);
  return 0;
}

The program should produce the following output,

The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
The sample mean is 16.54
The estimated variance is 4.2984
The largest value is 18.3
The smallest value is 12.6

Here is an example using sorted data,

#include <stdio.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_statistics.h>
int
main(void)
{
  double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
  double median, upperq, lowerq;
  printf ("Original dataset:  %g, %g, %g, %g, %g\n",
         data[0], data[1], data[2], data[3], data[4]);
  gsl_sort (data, 1, 5);
  printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
         data[0], data[1], data[2], data[3], data[4]);
  median
    = gsl_stats_median_from_sorted_data (data,
                                         1, 5);
  upperq
    = gsl_stats_quantile_from_sorted_data (data,
                                           1, 5,
                                           0.75);
  lowerq
    = gsl_stats_quantile_from_sorted_data (data,
                                           1, 5,
                                           0.25);
  printf ("The median is %g\n", median);
  printf ("The upper quartile is %g\n", upperq);
  printf ("The lower quartile is %g\n", lowerq);
  return 0;
}

This program should produce the following output,

Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6
Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
The median is 17.2
The upper quartile is 18.1
The lower quartile is 16.5

20.10 References and Further Reading

The standard reference for almost any topic in statistics is the multi-volume Advanced Theory of Statistics by Kendall and Stuart.

Maurice Kendall, Alan Stuart, and J. Keith Ord. The Advanced Theory of Statistics (multiple volumes) reprinted as Kendall’s Advanced Theory of Statistics. Wiley, ISBN 047023380X.

Many statistical concepts can be more easily understood by a Bayesian approach. The following book by Gelman, Carlin, Stern and Rubin gives a comprehensive coverage of the subject.

Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin. Bayesian Data Analysis. Chapman & Hall, ISBN 0412039915.

For physicists the Particle Data Group provides useful reviews of Probability and Statistics in the “Mathematical Tools” section of its Annual Review of Particle Physics.

Review of Particle Properties R.M. Barnett et al., Physical Review D54, 1 (1996)

The Review of Particle Physics is available online at the website http://pdg.lbl.gov/.

21 Histograms

This chapter describes functions for creating histograms. Histograms provide a convenient way of summarizing the distribution of a set of data. A histogram consists of a set of bins which count the number of events falling into a given range of a continuous variable x. In GSL the bins of a histogram contain floating-point numbers, so they can be used to record both integer and non-integer distributions. The bins can use arbitrary sets of ranges (uniformly spaced bins are the default). Both one and two-dimensional histograms are supported.

Once a histogram has been created it can also be converted into a probability distribution function. The library provides efficient routines for selecting random samples from probability distributions. This can be useful for generating simulations based on real data.

The functions are declared in the header files ‘gsl_histogram.h’ and

‘gsl_histogram2d.h’.

21.1 The histogram struct

A histogram is defined by the following struct,

[Data Type] gsl_histogram

size_t n This is the number of histogram bins

     double * range

The ranges of the bins are stored in an array of n + 1 elements pointed to by range.

     double * bin

The counts for each bin are stored in an array of n elements pointed to by bin. The bins are floating-point numbers, so you can increment them by non-integer values if necessary.

The range for bin[i] is given by range[i] to range[i+1]. For n bins there are n + 1 entries in the array range. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins are defined by the following inequality,

bin[i] corresponds to range[i] ≤ x < range[i+1]

Here is a diagram of the correspondence between ranges and bins on the number-line for x,

           [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
        ---|---------|---------|---------|---------|---------|---  x
         r[0]      r[1]      r[2]      r[3]      r[4]      r[5]

In this picture the values of the range array are denoted by r. On the left-hand side of each bin the square bracket ‘[’ denotes an inclusive lower bound (r ≤ x), and the round parentheses ‘)’ on the right-hand side denote an exclusive upper bound (x < r). Thus any samples which fall on the upper end of the histogram are excluded. If you want to include this value for the last bin you will need to add an extra bin to your histogram.

The gsl_histogram struct and its associated functions are defined in the header file

‘gsl_histogram.h’.

21.2 Histogram allocation

The functions for allocating memory to a histogram follow the style of malloc and free. In addition they also perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions call the error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn’t necessary to check every histogram alloc.

[Function] gsl_histogram * gsl_histogram_alloc (size t n)

This function allocates memory for a histogram with n bins, and returns a pointer to a newly created gsl_histogram struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM. The bins and ranges are not initialized, and should be prepared using one of the rangesetting functions below in order to make the histogram ready for use.

[Function] int gsl_histogram_set_ranges (gsl histogram * h, const double

   range[], size t size)

This function sets the ranges of the existing histogram h using the array range of size size. The values of the histogram bins are reset to zero. The range array should contain the desired bin limits. The ranges can be arbitrary, subject to the restriction that they are monotonically increasing.

The following example shows how to create a histogram with logarithmic bins with ranges [1,10), [10,100) and [100,1000).

     gsl_histogram * h = gsl_histogram_alloc (3);
     /* bin[0] covers the range 1 <= x < 10 */
     /* bin[1] covers the range 10 <= x < 100 */
     /* bin[2] covers the range 100 <= x < 1000 */
     double range[4] = { 1.0, 10.0, 100.0, 1000.0 };
     gsl_histogram_set_ranges (h, range, 4);

Note that the size of the range array should be defined to be one element bigger than the number of bins. The additional element is required for the upper value of the final bin.

[Function] int gsl_histogram_set_ranges_uniform (gsl histogram * h, double
   xmin, double xmax)

This function sets the ranges of the existing histogram h to cover the range xmin to xmax uniformly. The values of the histogram bins are reset to zero. The bin ranges are shown in the table below,

bin[0] corresponds to  xmin ≤ x < xmin + d
bin[1] corresponds to xmin + d ≤ x < xmin + 2d
… … …
bin[n-1] corresponds to xmin + (n − 1)d ≤ x < xmax

where d is the bin spacing, d = (xmax − xmin)/n.

[Function] void gsl_histogram_free (gsl histogram * h)

This function frees the histogram h and all of the memory associated with it.

21.3 Copying Histograms

[Function] int gsl_histogram_memcpy (gsl histogram * dest, const gsl histogram

   * src)

This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.

[Function] gsl_histogram * gsl_histogram_clone (const gsl histogram * src)

This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

21.4 Updating and accessing histogram elements

There are two ways to access histogram bins, either by specifying an x coordinate or by using the bin-index directly. The functions for accessing the histogram through x coordinates use a binary search to identify the bin which covers the appropriate range.

[Function] int gsl_histogram_increment (gsl histogram * h, double x)

This function updates the histogram h by adding one (1.0) to the bin whose range contains the coordinate x. If x lies in the valid range of the histogram then the function returns zero to indicate success. If x is less than the lower limit of the histogram then the function returns GSL_EDOM, and none of bins are modified. Similarly, if the value of x is greater than or equal to the upper limit of the histogram then the function returns GSL_EDOM, and none of the bins are modified. The error handler is not called, however, since it is often necessary to compute histograms for a small range of a larger dataset, ignoring the values outside the range of interest.

[Function] int gsl_histogram_accumulate (gsl histogram * h, double x, double

   weight)

This function is similar to gsl_histogram_increment but increases the value of the appropriate bin in the histogram h by the floating-point number weight.

[Function] double gsl_histogram_get (const gsl histogram * h, size t i)

This function returns the contents of the i-th bin of the histogram h. If i lies outside the valid range of indices for the histogram then the error handler is called with an error code of GSL_EDOM and the function returns 0.

[Function] int gsl_histogram_get_range (const gsl histogram * h, size t i,

double * lower, double * upper) This function finds the upper and lower range limits of the i-th bin of the histogram h. If the index i is valid then the corresponding range limits are stored in lower and upper. The lower limit is inclusive (i.e. events with this coordinate are included in the bin) and the upper limit is exclusive (i.e. events with the coordinate of the upper limit are excluded and fall in the neighboring higher bin, if it exists). The function returns 0 to indicate success. If i lies outside the valid range of indices for the histogram then the error handler is called and the function returns an error code of GSL_EDOM.

[Function] double gsl_histogram_max (const gsl histogram * h)

[Function] double gsl_histogram_min (const gsl histogram * h)

[Function] size_t gsl_histogram_bins (const gsl histogram * h)

These functions return the maximum upper and minimum lower range limits and the number of bins of the histogram h. They provide a way of determining these values without accessing the gsl_histogram struct directly.

[Function] void gsl_histogram_reset (gsl histogram * h)

This function resets all the bins in the histogram h to zero.

21.5 Searching histogram ranges

The following functions are used by the access and update routines to locate the bin which corresponds to a given x coordinate.

[Function] int gsl_histogram_find (const gsl histogram * h, double x, size t * i)

This function finds and sets the index i to the bin number which covers the coordinate x in the histogram h. The bin is located using a binary search. The search includes an optimization for histograms with uniform range, and will return the correct bin immediately in this case. If x is found in the range of the histogram then the function sets the index i and returns GSL_SUCCESS. If x lies outside the valid range of the histogram then the function returns GSL_EDOM and the error handler is invoked.

21.6 Histogram Statistics

[Function] double gsl_histogram_max_val (const gsl histogram * h)

This function returns the maximum value contained in the histogram bins.

[Function] size_t gsl_histogram_max_bin (const gsl histogram * h)

This function returns the index of the bin containing the maximum value. In the case where several bins contain the same maximum value the smallest index is returned.

[Function] double gsl_histogram_min_val (const gsl histogram * h)

This function returns the minimum value contained in the histogram bins.

[Function] size_t gsl_histogram_min_bin (const gsl histogram * h)

This function returns the index of the bin containing the minimum value. In the case where several bins contain the same maximum value the smallest index is returned.

[Function] double gsl_histogram_mean (const gsl histogram * h)

This function returns the mean of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.

[Function] double gsl_histogram_sigma (const gsl histogram * h)

This function returns the standard deviation of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.

[Function] double gsl_histogram_sum (const gsl histogram * h)

This function returns the sum of all bin values. Negative bin values are included in the sum.

21.7 Histogram Operations

[Function] int gsl_histogram_equal_bins_p (const gsl histogram * h1, const

gsl histogram * h2) This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.

[Function] int gsl_histogram_add (gsl histogram * h1, const gsl histogram * h2)

This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h′_{1}(i) = h_{1}(i) + h_{2}(i). The two histograms must have identical bin ranges.

[Function] int gsl_histogram_sub (gsl histogram * h1, const gsl histogram * h2)

This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h′_{1}(i) = h_{1}(i) − h_{2}(i). The two histograms must have identical bin ranges.

[Function] int gsl_histogram_mul (gsl histogram * h1, const gsl histogram * h2)

This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h′_{1}(i) = h_{1}(i) ∗ h_{2}(i). The two histograms must have identical bin ranges.

[Function] int gsl_histogram_div (gsl histogram * h1, const gsl histogram * h2)

This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h′_{1}(i) = h_{1}(i)/h_{2}(i). The two histograms must have identical bin ranges.

[Function] int gsl_histogram_scale (gsl histogram * h, double scale)

This function multiplies the contents of the bins of histogram h by the constant scale, i.e. h′_{1}(i) = h_{1}(i) ∗ scale.

[Function] int gsl_histogram_shift (gsl histogram * h, double offset)

This function shifts the contents of the bins of histogram h by the constant offset, i.e. h′_{1}(i) = h_{1}(i) + offset.

21.8 Reading and writing histograms

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

[Function] int gsl_histogram_fwrite (FILE * stream, const gsl histogram * h)

This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

[Function] int gsl_histogram_fread (FILE * stream, gsl histogram * h)

This function reads into the histogram h from the open stream stream in binary format. The histogram h must be preallocated with the correct size since the function uses the number of bins in h to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

[Function] int gsl_histogram_fprintf (FILE * stream, const gsl histogram * h,

   const char * range_format, const char * bin_format)

This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format specifiers range format and bin format. These should be one of the %g, %e or %f formats for floating point numbers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file. The histogram output is formatted in three columns, and the columns are separated by spaces, like this,

     range[0] range[1] bin[0]
     range[1] range[2] bin[1]
     range[2] range[3] bin[2]
     ....
     range[n-1] range[n] bin[n-1]

The values of the ranges are formatted using range format and the value of the bins are formatted using bin format. Each line contains the lower and upper limit of the range of the bins and the value of the bin itself. Since the upper limit of one bin is the lower limit of the next there is duplication of these values between lines but this allows the histogram to be manipulated with line-oriented tools.

[Function] int gsl_histogram_fscanf (FILE * stream, gsl histogram * h)

This function reads formatted data from the stream stream into the histogram h. The data is assumed to be in the three-column format used by gsl_histogram_fprintf. The histogram h must be preallocated with the correct length since the function uses the size of h to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

21.9 Resampling from histograms

A histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where the value of x falls in the range of that bin. The probability distribution function has the one-dimensional form p(x)dx where,

p(x) = n_{i}/(Nw_{i})

In this equation n_{i} is the number of events in the bin which contains x, w_{i} is the width of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.

21.10 The histogram probability distribution struct

The probability distribution function for a histogram consists of a set of bins which measure the probability of an event falling into a given range of a continuous variable x. A probability distribution function is defined by the following struct, which actually stores the cumulative probability distribution function. This is the natural quantity for generating samples via the inverse transform method, because there is a one-to-one mapping between the cumulative probability distribution and the range [0,1]. It can be shown that by taking a uniform random number in this range and finding its corresponding coordinate in the cumulative probability distribution we obtain samples with the desired probability distribution.

[Data Type] gsl_histogram_pdf

size_t n This is the number of bins used to approximate the probability distribution function.

double * range

The ranges of the bins are stored in an array of n + 1 elements pointed to by range.

double * sum

The cumulative probability for the bins is stored in an array of n elements pointed to by sum.

The following functions allow you to create a gsl_histogram_pdf struct which represents this probability distribution and generate random samples from it.

[Function] gsl_histogram_pdf * gsl_histogram_pdf_alloc (size t n)

This function allocates memory for a probability distribution with n bins and returns a pointer to a newly initialized gsl_histogram_pdf struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error

code of GSL_ENOMEM.

[Function] int gsl_histogram_pdf_init (gsl histogram pdf * p, const

gsl histogram * h) This function initializes the probability distribution p with the contents of the histogram h. If any of the bins of h are negative then the error handler is invoked with an error code of GSL_EDOM because a probability distribution cannot contain negative values.

[Function] void gsl_histogram_pdf_free (gsl histogram pdf * p)

This function frees the probability distribution function p and all of the memory associated with it.

[Function] double gsl_histogram_pdf_sample (const gsl histogram pdf * p,

double r) This function uses r, a uniform random number between zero and one, to compute a single random sample from the probability distribution p. The algorithm used to compute the sample s is given by the following formula,

s = range[i] + δ ∗ (range[i + 1] − range[i])

where i is the index which satisfies sum[i] ≤ r < sum[i + 1] and delta is (r − sum[i])/(sum[i + 1] − sum[i]).

21.11 Example programs for histograms

The following program shows how to make a simple histogram of a column of numerical data supplied on stdin. The program takes three arguments, specifying the upper and lower bounds of the histogram and the number of bins. It then reads numbers from stdin, one line at a time, and adds them to the histogram. When there is no more data to read it prints out the accumulated histogram using gsl_histogram_fprintf.

     #include <stdio.h>
     #include <stdlib.h>
     #include <gsl/gsl_histogram.h>
     int
     main (int argc, char **argv)
     {
       double a, b;
       size_t n;
       if (argc != 4)
         {
           printf ("Usage: gsl-histogram xmin xmax n\n"
                   "Computes a histogram of the data "
                   "on stdin using n bins from xmin "
                   "to xmax\n");
           exit (0);
         }
       a = atof (argv[1]);
       b = atof (argv[2]);
       n = atoi (argv[3]);
       {
         double x;
         gsl_histogram * h = gsl_histogram_alloc (n);
         gsl_histogram_set_ranges_uniform (h, a, b);
         while (fscanf (stdin, "%lg", &x) == 1)
           {
             gsl_histogram_increment (h, x);
           }
         gsl_histogram_fprintf (stdout, h, "%g", "%g");
         gsl_histogram_free (h);
       }
       exit (0);
     }

Here is an example of the program in use. We generate 10000 random samples from a Cauchy distribution with a width of 30 and histogram them over the range -100 to 100, using 200 bins.

     $ gsl-randist 0 10000 cauchy 30
        | gsl-histogram -100 100 200 > histogram.dat

A plot of the resulting histogram shows the familiar shape of the Cauchy distribution and the fluctuations caused by the finite sample size.

     $ awk '{print $1, $3 ; print $2, $3}' histogram.dat
        | graph -T X

21.12 Two dimensional histograms

A two dimensional histogram consists of a set of bins which count the number of events falling in a given area of the (x, y) plane. The simplest way to use a two dimensional histogram is to record two-dimensional position information, n(x, y). Another possibility is to form a joint distribution by recording related variables. For example a detector might record both the position of an event (x) and the amount of energy it deposited E. These could be histogrammed as the joint distribution n(x, E).

21.13 The 2D histogram struct

Two dimensional histograms are defined by the following struct,

[Data Type] gsl_histogram2d
     size_t nx, ny

This is the number of histogram bins in the x and y directions.

     double * xrange

The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.

     double * yrange

The ranges of the bins in the y-direction are stored in an array of ny + 1 elements pointed to by yrange.

     double * bin

The counts for each bin are stored in an array pointed to by bin. The bins are floating-point numbers, so you can increment them by non-integer values if necessary. The array bin stores the two dimensional array of bins in a single block of memory according to the mapping bin(i,j) =

               bin[i * ny + j].

The range for bin(i,j) is given by xrange[i] to xrange[i+1] in the x-direction and yrange[j] to yrange[j+1] in the y-direction. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins are defined by the following inequality,

bin(i,j) corresponds to xrange[i] ≤ x < xrange[i + 1]
and  yrange[j] ≤ y < yrange[j + 1]

Note that any samples which fall on the upper sides of the histogram are excluded. If you want to include these values for the side bins you will need to add an extra row or column to your histogram.

The gsl_histogram2d struct and its associated functions are defined in the header file

‘gsl_histogram2d.h’.

21.14 2D Histogram allocation

The functions for allocating memory to a 2D histogram follow the style of malloc and free. In addition they also perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions call the error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn’t necessary to check every 2D histogram

alloc.
[Function] gsl_histogram2d * gsl_histogram2d_alloc (size t nx, size t ny)

This function allocates memory for a two-dimensional histogram with nx bins in the x direction and ny bins in the y direction. The function returns a pointer to a newly created gsl_histogram2d struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM. The bins and ranges must be initialized with one of the functions below before the histogram is ready for use.

[Function] int gsl_histogram2d_set_ranges (gsl histogram2d * h, const double

xrange[], size t xsize, const double yrange[], size t ysize)

This function sets the ranges of the existing histogram h using the arrays xrange and yrange of size xsize and ysize respectively. The values of the histogram bins are reset to zero.

[Function] int gsl_histogram2d_set_ranges_uniform (gsl histogram2d * h,

double xmin, double xmax, double ymin, double ymax) This function sets the ranges of the existing histogram h to cover the ranges xmin to xmax and ymin to ymax uniformly. The values of the histogram bins are reset to zero.

[Function] void gsl_histogram2d_free (gsl histogram2d * h)

This function frees the 2D histogram h and all of the memory associated with it.

21.15 Copying 2D Histograms

[Function] int gsl_histogram2d_memcpy (gsl histogram2d * dest, const

gsl histogram2d * src) This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.

[Function] gsl_histogram2d * gsl_histogram2d_clone (const gsl histogram2d
   * src)

This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

21.16 Updating and accessing 2D histogram elements

You can access the bins of a two-dimensional histogram either by specifying a pair of (x, y) coordinates or by using the bin indices (i, j) directly. The functions for accessing the histogram through (x, y) coordinates use binary searches in the x and y directions to identify the bin which covers the appropriate range.

[Function] int gsl_histogram2d_increment (gsl histogram2d * h, double x,

double y) This function updates the histogram h by adding one (1.0) to the bin whose x and y ranges contain the coordinates (x,y). If the point (x, y) lies inside the valid ranges of the histogram then the function returns zero to indicate success. If (x, y) lies outside the limits of the histogram then the function returns GSL_EDOM, and none of the bins are modified. The error handler is not called, since it is often necessary to compute histograms for a small range of a larger dataset, ignoring any coordinates outside the range of interest.

[Function] int gsl_histogram2d_accumulate (gsl histogram2d * h, double x,

double y, double weight) This function is similar to gsl_histogram2d_increment but increases the value of the appropriate bin in the histogram h by the floating-point number weight.

[Function] double gsl_histogram2d_get (const gsl histogram2d * h, size t i,

size t j) This function returns the contents of the (i,j)-th bin of the histogram h. If (i,j) lies outside the valid range of indices for the histogram then the error handler is called with an error code of GSL_EDOM and the function returns 0.

[Function] int gsl_histogram2d_get_xrange (const gsl histogram2d * h, size t

i, double * xlower, double * xupper)

[Function] int gsl_histogram2d_get_yrange (const gsl histogram2d * h, size t

j, double * ylower, double * yupper) These functions find the upper and lower range limits of the i-th and j-th bins in the x and y directions of the histogram h. The range limits are stored in xlower and xupper or ylower and yupper. The lower limits are inclusive (i.e. events with these coordinates are included in the bin) and the upper limits are exclusive (i.e. events with the value of the upper limit are not included and fall in the neighboring higher bin, if it exists). The functions return 0 to indicate success. If i or j lies outside the valid range of indices for the histogram then the error handler is called with an error

code of GSL_EDOM.

[Function] double gsl_histogram2d_xmax (const gsl histogram2d * h)

[Function] double gsl_histogram2d_xmin (const gsl histogram2d * h)

[Function] size_t gsl_histogram2d_nx (const gsl histogram2d * h)

[Function] double gsl_histogram2d_ymax (const gsl histogram2d * h)

[Function] double gsl_histogram2d_ymin (const gsl histogram2d * h)

[Function] size_t gsl_histogram2d_ny (const gsl histogram2d * h)

These functions return the maximum upper and minimum lower range limits and the number of bins for the x and y directions of the histogram h. They provide a way of determining these values without accessing the gsl_histogram2d struct directly.

[Function] void gsl_histogram2d_reset (gsl histogram2d * h)

This function resets all the bins of the histogram h to zero.

21.17 Searching 2D histogram ranges

The following functions are used by the access and update routines to locate the bin which corresponds to a given (x, y) coordinate.

[Function] int gsl_histogram2d_find (const gsl histogram2d * h, double x,

double y, size t * i, size t * j) This function finds and sets the indices i and j to the to the bin which covers the coordinates (x,y). The bin is located using a binary search. The search includes an optimization for histograms with uniform ranges, and will return the correct bin immediately in this case. If (x, y) is found then the function sets the indices (i,j) and returns GSL_SUCCESS. If (x, y) lies outside the valid range of the histogram then the function returns GSL_EDOM and the error handler is invoked.

21.18 2D Histogram Statistics

[Function] double gsl_histogram2d_max_val (const gsl histogram2d * h)

This function returns the maximum value contained in the histogram bins.

[Function] void gsl_histogram2d_max_bin (const gsl histogram2d * h, size t * i,

size t * j) This function finds the indices of the bin containing the maximum value in the histogram h and stores the result in (i,j). In the case where several bins contain the same maximum value the first bin found is returned.

[Function] double gsl_histogram2d_min_val (const gsl histogram2d * h)

This function returns the minimum value contained in the histogram bins.

[Function] void gsl_histogram2d_min_bin (const gsl histogram2d * h, size t * i,

size t * j) This function finds the indices of the bin containing the minimum value in the histogram h and stores the result in (i,j). In the case where several bins contain the same maximum value the first bin found is returned.

[Function] double gsl_histogram2d_xmean (const gsl histogram2d * h)

This function returns the mean of the histogrammed x variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.

[Function] double gsl_histogram2d_ymean (const gsl histogram2d * h)

This function returns the mean of the histogrammed y variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.

[Function] double gsl_histogram2d_xsigma (const gsl histogram2d * h)

This function returns the standard deviation of the histogrammed x variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.

[Function] double gsl_histogram2d_ysigma (const gsl histogram2d * h)

This function returns the standard deviation of the histogrammed y variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.

[Function] double gsl_histogram2d_cov (const gsl histogram2d * h)

This function returns the covariance of the histogrammed x and y variables, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.

[Function] double gsl_histogram2d_sum (const gsl histogram2d * h)

This function returns the sum of all bin values. Negative bin values are included in the sum.

21.19 2D Histogram Operations

[Function] int gsl_histogram2d_equal_bins_p (const gsl histogram2d * h1,

const gsl histogram2d * h2) This function returns 1 if all the individual bin ranges of the two histograms are identical, and 0 otherwise.

[Function] int gsl_histogram2d_add (gsl histogram2d * h1, const

gsl histogram2d * h2) This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h′_{1}(i, j) = h_{1}(i, j) + h_{2}(i, j). The two histograms must have identical bin ranges.

[Function] int gsl_histogram2d_sub (gsl histogram2d * h1, const

gsl histogram2d * h2) This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h′_{1}(i, j) = h_{1}(i, j) − h_{2}(i, j). The two histograms must have identical bin ranges.

[Function] int gsl_histogram2d_mul (gsl histogram2d * h1, const

gsl histogram2d * h2) This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h′_{1}(i, j) = h_{1}(i, j) ∗ h_{2}(i, j). The two histograms must have identical bin ranges.

[Function] int gsl_histogram2d_div (gsl histogram2d * h1, const

gsl histogram2d * h2) This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h′_{1}(i, j) = h_{1}(i, j)/h_{2}(i, j). The two histograms must have identical bin ranges.

[Function] int gsl_histogram2d_scale (gsl histogram2d * h, double scale)

This function multiplies the contents of the bins of histogram h by the constant scale, i.e. h′_{1}(i, j) = h_{1}(i, j) ∗ scale.

[Function] int gsl_histogram2d_shift (gsl histogram2d * h, double offset)

This function shifts the contents of the bins of histogram h by the constant offset, i.e. h′_{1}(i, j) = h_{1}(i, j) + offset.

21.20 Reading and writing 2D histograms

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

[Function] int gsl_histogram2d_fwrite (FILE * stream, const gsl histogram2d

* h) This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

[Function] int gsl_histogram2d_fread (FILE * stream, gsl histogram2d * h)

This function reads into the histogram h from the stream stream in binary format. The histogram h must be preallocated with the correct size since the function uses the number of x and y bins in h to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

[Function] int gsl_histogram2d_fprintf (FILE * stream, const gsl histogram2d

* h, const char * range_format, const char * bin_format) This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format specifiers range format and bin format. These should be one of the %g, %e or %f formats for floating point numbers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file. The histogram output is formatted in five columns, and the columns are separated by spaces, like this,

      xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
      xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
      xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
      ....
      xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)
      xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
      xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
      xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
      ....
      xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)
      ....
      xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
      xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
      xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
      ....
      xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)

Each line contains the lower and upper limits of the bin and the contents of the bin. Since the upper limits of the each bin are the lower limits of the neighboring bins there is duplication of these values but this allows the histogram to be manipulated with line-oriented tools.

[Function] int gsl_histogram2d_fscanf (FILE * stream, gsl histogram2d * h)

This function reads formatted data from the stream stream into the histogram h. The data is assumed to be in the five-column format used by gsl_histogram_fprintf. The histogram h must be preallocated with the correct lengths since the function uses the sizes of h to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

21.21 Resampling from 2D histograms

As in the one-dimensional case, a two-dimensional histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where (x,y) falls in the range of that bin. For a two-dimensional histogram the probability distribution takes the form p(x, y)dxdy where,

p(x, y) = n_{ij}/(NA_{ij})

In this equation n_{ij} is the number of events in the bin which contains (x, y), A_{ij} is the area of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.

[Data Type] gsl_histogram2d_pdf
size_t nx, ny

This is the number of histogram bins used to approximate the probability distribution function in the x and y directions.

double * xrange

The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.

double * yrange

The ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to by yrange.

double * sum

The cumulative probability for the bins is stored in an array of nx*ny elements pointed to by sum.

The following functions allow you to create a gsl_histogram2d_pdf struct which represents a two dimensional probability distribution and generate random samples from it.

[Function] gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc (size t nx,

size t ny) This function allocates memory for a two-dimensional probability distribution of size nx-by-ny and returns a pointer to a newly initialized gsl_histogram2d_pdf struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] int gsl_histogram2d_pdf_init (gsl histogram2d pdf * p, const

gsl histogram2d * h) This function initializes the two-dimensional probability distribution calculated p from the histogram h. If any of the bins of h are negative then the error handler is invoked with an error code of GSL_EDOM because a probability distribution cannot contain negative values.

[Function] void gsl_histogram2d_pdf_free (gsl histogram2d pdf * p)

This function frees the two-dimensional probability distribution function p and all of the memory associated with it.

[Function] int gsl_histogram2d_pdf_sample (const gsl histogram2d pdf * p,

double r1, double r2, double * x, double * y) This function uses two uniform random numbers between zero and one, r1 and r2, to compute a single random sample from the two-dimensional probability distribution p.

21.22 Example programs for 2D histograms

This program demonstrates two features of two-dimensional histograms. First a 10-by-10 two-dimensional histogram is created with x and y running from 0 to 1. Then a few sample points are added to the histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height of 5 and at (0.7,0.9) with a height of 0.5. This histogram with three events is used to generate a random sample of 1000 simulated events, which are printed out.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_histogram2d.h>
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);
gsl_histogram2d_set_ranges_uniform (h,
                                    0.0, 1.0,
                                    0.0, 1.0);
gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
{
  int i;
  gsl_histogram2d_pdf * p
    = gsl_histogram2d_pdf_alloc (h->nx, h->ny);
  gsl_histogram2d_pdf_init (p, h);
  for (i = 0; i < 1000; i++) {
    double x, y;
    double u = gsl_rng_uniform (r);
    double v = gsl_rng_uniform (r);
    gsl_histogram2d_pdf_sample (p, u, v, &x, &y);
    printf ("%g %g\n", x, y);

         }
       }
      return 0;
     }

The following plot shows the distribution of the simulated events. Using a higher resolution grid we can see the original underlying histogram and also the statistical fluctuations caused by the events being uniformly distributed over the area of the original bins.

22 N-tuples

This chapter describes functions for creating and manipulating ntuples, sets of values associated with events. The ntuples are stored in files. Their values can be extracted in any combination and booked in a histogram using a selection function.

The values to be stored are held in a user-defined data structure, and an ntuple is created associating this data structure with a file. The values are then written to the file (normally inside a loop) using the ntuple functions described below.

A histogram can be created from ntuple data by providing a selection function and a value function. The selection function specifies whether an event should be included in the subset to be analyzed or not. The value function computes the entry to be added to the histogram for each event.

All the ntuple functions are defined in the header file ‘gsl_ntuple.h’

22.1 The ntuple struct

Ntuples are manipulated using the gsl_ntuple struct. This struct contains information on the file where the ntuple data is stored, a pointer to the current ntuple data row and the size of the user-defined ntuple data struct.

     typedef struct {
         FILE * file;
         void * ntuple_data;
         size_t size;
     } gsl_ntuple;

22.2 Creating ntuples

[Function] gsl_ntuple * gsl_ntuple_create (char * filename, void *
        ntuple_data, size t size)

This function creates a new write-only ntuple file filename for ntuples of size size and returns a pointer to the newly created ntuple struct. Any existing file with the same name is truncated to zero length and overwritten. A pointer to memory for the current ntuple row ntuple data must be supplied—this is used to copy ntuples in and out of the file.

22.3 Opening an existing ntuple file

[Function] gsl_ntuple * gsl_ntuple_open (char * filename, void *
        ntuple_data, size t size)

This function opens an existing ntuple file filename for reading and returns a pointer to a corresponding ntuple struct. The ntuples in the file must have size size. A pointer to memory for the current ntuple row ntuple data must be supplied—this is used to copy ntuples in and out of the file.

22.4 Writing ntuples

[Function] int gsl_ntuple_write (gsl ntuple * ntuple)

This function writes the current ntuple ntuple->ntuple data of size ntuple->size to the corresponding file.

[Function] int gsl_ntuple_bookdata (gsl ntuple * ntuple)

This function is a synonym for gsl_ntuple_write.

22.5 Reading ntuples

[Function] int gsl_ntuple_read (gsl ntuple * ntuple)

This function reads the current row of the ntuple file for ntuple and stores the values in ntuple->data.

22.6 Closing an ntuple file

[Function] int gsl_ntuple_close (gsl ntuple * ntuple)

This function closes the ntuple file ntuple and frees its associated allocated memory.

22.7 Histogramming ntuple values

Once an ntuple has been created its contents can be histogrammed in various ways using the function gsl_ntuple_project. Two user-defined functions must be provided, a function to select events and a function to compute scalar values. The selection function and the value function both accept the ntuple row as a first argument and other parameters as a second argument.

The selection function determines which ntuple rows are selected for histogramming. It is defined by the following struct,

      typedef struct {
        int (* function) (void * ntuple_data, void * params);
        void * params;
      } gsl_ntuple_select_fn;

The struct component function should return a non-zero value for each ntuple row that is to be included in the histogram.

The value function computes scalar values for those ntuple rows selected by the selection function,

      typedef struct {
        double (* function) (void * ntuple_data, void * params);
        void * params;
      } gsl_ntuple_value_fn;

In this case the struct component function should return the value to be added to the histogram for the ntuple row.

[Function] int gsl_ntuple_project (gsl histogram * h, gsl ntuple * ntuple,

gsl ntuple value fn * value_func, gsl ntuple select fn * select_func) This function updates the histogram h from the ntuple ntuple using the functions value func and select func. For each ntuple row where the selection function select func is non-zero the corresponding value of that row is computed using the function value func and added to the histogram. Those ntuple rows where select func returns zero are ignored. New entries are added to the histogram, so subsequent calls can be used to accumulate further data in the same histogram.

22.8 Examples

The following example programs demonstrate the use of ntuples in managing a large dataset. The first program creates a set of 10,000 simulated “events”, each with 3 associated values (x, y, z). These are generated from a gaussian distribution with unit variance, for demonstration purposes, and written to the ntuple file ‘test.dat’.

#include <gsl/gsl_ntuple.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
struct data
{
double x;
double y;
double z;
};
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
struct data ntuple_row;
int i;
gsl_ntuple *ntuple
  = gsl_ntuple_create ("test.dat", &ntuple_row,
                       sizeof (ntuple_row));
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < 10000; i++)
  {
    ntuple_row.x = gsl_ran_ugaussian (r);
    ntuple_row.y = gsl_ran_ugaussian (r);
    ntuple_row.z = gsl_ran_ugaussian (r);
    gsl_ntuple_write (ntuple);
  }

gsl_ntuple_close (ntuple);
return 0;
}

The next program analyses the ntuple data in the file ‘test.dat’. The analysis procedure is to compute the squared-magnitude of each event, E^{2} = x^{2} +y^{2} +z^{2}, and select only those which exceed a lower limit of 1.5. The selected events are then histogrammed using their E^{2} values.

#include <math.h>
#include <gsl/gsl_ntuple.h>
#include <gsl/gsl_histogram.h>
struct data
{
double x;
double y;
double z;
};
int sel_func (void *ntuple_data, void *params);
double val_func (void *ntuple_data, void *params);
int
main (void)
{
struct data ntuple_row;
gsl_ntuple *ntuple
  = gsl_ntuple_open ("test.dat", &ntuple_row,
                     sizeof (ntuple_row));
double lower = 1.5;
gsl_ntuple_select_fn S;
gsl_ntuple_value_fn V;
gsl_histogram *h = gsl_histogram_alloc (100);
gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);
S.function = &sel_func;
S.params = &lower;
V.function = &val_func;
V.params = 0;
gsl_ntuple_project (h, ntuple, &V, &S);
gsl_histogram_fprintf (stdout, h, "%f", "%f");
gsl_histogram_free (h);

gsl_ntuple_close (ntuple);
return 0;
}
int
sel_func (void *ntuple_data, void *params)
{
struct data * data = (struct data *) ntuple_data;
double x, y, z, E2, scale;
scale = *(double *) params;
x = data->x;
y = data->y;
z = data->z;
E2 = x * x + y * y + z * z;
return E2 > scale;
}
double
val_func (void *ntuple_data, void *params)
{
struct data * data = (struct data *) ntuple_data;
double x, y, z;
x = data->x;
y = data->y;
z = data->z;
return x * x + y * y + z * z;
}

The following plot shows the distribution of the selected events. Note the cut-off at the lower bound.

22.9 References and Further Reading

Further information on the use of ntuples can be found in the documentation for the cern packages paw and hbook (available online).

23 Monte Carlo Integration

This chapter describes routines for multidimensional Monte Carlo integration. These include the traditional Monte Carlo method and adaptive algorithms such as vegas and miser which use importance sampling and stratified sampling techniques. Each algorithm computes an estimate of a multidimensional definite integral of the form,

I = ∫_{x_{l}}^{x_{u}} dx ∫_{y_{l}}^{y_{u}} dy ...f(x, y, ...)

over a hypercubic region ((x_{l}, x_{u}), (y_{l}, y_{u}), ...) using a fixed number of function calls. The routines also provide a statistical estimate of the error on the result. This error estimate should be taken as a guide rather than as a strict error bound—random sampling of the region may not uncover all the important features of the function, resulting in an underestimate of the error.

The functions are defined in separate header files for each routine, gsl_monte_plain.h,

‘gsl_monte_miser.h’ and ‘gsl_monte_vegas.h’.

23.1 Interface

All of the Monte Carlo integration routines use the same general form of interface. There is an allocator to allocate memory for control variables and workspace, a routine to initialize those control variables, the integrator itself, and a function to free the space when done.

Each integration function requires a random number generator to be supplied, and returns an estimate of the integral and its standard deviation. The accuracy of the result is determined by the number of function calls specified by the user. If a known level of accuracy is required this can be achieved by calling the integrator several times and averaging the individual results until the desired accuracy is obtained.

Random sample points used within the Monte Carlo routines are always chosen strictly within the integration region, so that endpoint singularities are automatically avoided.

The function to be integrated has its own datatype, defined in the header file

‘gsl_monte.h’.
[Data Type] gsl_monte_function

This data type defines a general function with parameters for Monte Carlo integration.

     double (* f) (double * x, size_t dim, void * params)

this function should return the value f(x, params) for the argument x and parameters params, where x is an array of size dim giving the coordinates of the point where the function is to be evaluated.

     size_t dim

the number of dimensions for x.

     void * params

a pointer to the parameters of the function.

Here is an example for a quadratic function in two dimensions,

f(x, y) = ax^{2} + bxy + cy^{2}

with a = 3, b = 2, c = 1. The following code defines a gsl_monte_function F which you could pass to an integrator:

     struct my_f_params { double a; double b; double c; };
     double
     my_f (double x[], size_t dim, void * p) {
        struct my_f_params * fp = (struct my_f_params *)p;
        if (dim != 2)
           {
             fprintf (stderr, "error: dim != 2");
             abort ();
           }
        return  fp->a * x[0] * x[0]
                  + fp->b * x[0] * x[1]
                    + fp->c * x[1] * x[1];
     }
     gsl_monte_function F;
     struct my_f_params params = { 3.0, 2.0, 1.0 };
     F.f = &my_f;
     F.dim = 2;
     F.params = &params;

The function f(x) can be evaluated using the following macro,

     #define GSL_MONTE_FN_EVAL(F,x)
         (*((F)->f))(x,(F)->dim,(F)->params)

23.2 PLAIN Monte Carlo

The plain Monte Carlo algorithm samples points randomly from the integration region to estimate the integral and its error. Using this algorithm the estimate of the integral E(f; N) for N randomly distributed points x_{i} is given by,

E(f; N) = V ⟨f⟩ = \frac{V}{N} ∑_{i}^{N} f(x_{i})

where V is the volume of the integration region. The error on this estimate σ(E; N) is calculated from the estimated variance of the mean,

σ^{2}(E; N) = \frac{V}{N} ∑_{i}^{N}(f(x_{i}) − ⟨f⟩)^{2}.

For large N this variance decreases asymptotically as Var(f)/N, where Var(f) is the true variance of the function over the integration region. The error estimate itself should decrease as σ(f)/\sqrt{N}. The familiar law of errors decreasing as 1/\sqrt{N} applies—to reduce the error by a factor of 10 requires a 100-fold increase in the number of sample points.

The functions described in this section are declared in the header file

‘gsl_monte_plain.h’.
[Function] gsl_monte_plain_state * gsl_monte_plain_alloc (size t dim)

This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions.

[Function] int gsl_monte_plain_init (gsl monte plain state* s)

This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

[Function] int gsl_monte_plain_integrate (gsl monte function * f, double *

xl, double * xu, size t dim, size t calls, gsl rng * r, gsl monte plain state * s, double * result, double * abserr)

This routines uses the plain Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.

[Function] void gsl_monte_plain_free (gsl monte plain state * s)

This function frees the memory associated with the integrator state s.

23.3 MISER

The miser algorithm of Press and Farrar is based on recursive stratified sampling. This technique aims to reduce the overall integration error by concentrating integration points in the regions of highest variance.

The idea of stratified sampling begins with the observation that for two disjoint regions a and b with Monte Carlo estimates of the integral E_{a}(f) and E_{b}(f) and variances σ_{a}^{2}(f) and σ_{b}^{2}(f), the variance Var(f) of the combined estimate E(f) = \frac{1}{2}(E_{a}(f) + E_{b}(f)) is given by,

Var(f) = \frac{σ_{a}^{2}(f)}{4N_{a}} + \frac{σ_{b}^{2}(f)}{4N_{b}}.

It can be shown that this variance is minimized by distributing the points such that,

\frac{N_{a}}{N_{a} + N_{b}} = \frac{σ_{a}}{σ_{a} + σ_{b}} .

Hence the smallest error estimate is obtained by allocating sample points in proportion to the standard deviation of the function in each sub-region.

The miser algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions at each step. The direction is chosen by examining all d possible bisections and selecting the one which will minimize the combined variance of the two subregions. The variance in the sub-regions is estimated by sampling with a fraction of the total number of points available to the current step. The same procedure is then repeated recursively for each of the two half-spaces from the best bisection. The remaining sample points are allocated to the sub-regions using the formula for N_{a} and N_{b}. This recursive allocation of integration points continues down to a user-specified depth where each subregion is integrated using a plain Monte Carlo estimate. These individual values and their error estimates are then combined upwards to give an overall result and an estimate of its error.

The functions described in this section are declared in the header file

‘gsl_monte_miser.h’.
[Function] gsl_monte_miser_state * gsl_monte_miser_alloc (size t dim)

This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.

[Function] int gsl_monte_miser_init (gsl monte miser state* s)

This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

[Function] int gsl_monte_miser_integrate (gsl monte function * f, double *

xl, double * xu, size t dim, size t calls, gsl rng * r, gsl monte miser state * s, double * result, double * abserr)

This routines uses the miser Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.

[Function] void gsl_monte_miser_free (gsl monte miser state * s)

This function frees the memory associated with the integrator state s.

The miser algorithm has several configurable parameters. The following variables can be accessed through the gsl_monte_miser_state struct,

[Variable] double estimate_frac

This parameter specifies the fraction of the currently available number of function calls which are allocated to estimating the variance at each recursive step. The default value is 0.1.

[Variable] size_t min_calls

This parameter specifies the minimum number of function calls required for each estimate of the variance. If the number of function calls allocated to the estimate using estimate frac falls below min calls then min calls are used instead. This ensures that each estimate maintains a reasonable level of accuracy. The default value of min calls

is 16 * dim.
[Variable] size_t min_calls_per_bisection

This parameter specifies the minimum number of function calls required to proceed with a bisection step. When a recursive step has fewer calls available than min calls per bisection it performs a plain Monte Carlo estimate of the current subregion and terminates its branch of the recursion. The default value of this parameter

is 32 * min_calls.

[Variable] double alpha

This parameter controls how the estimated variances for the two sub-regions of a bisection are combined when allocating points. With recursive sampling the overall variance should scale better than 1/N, since the values from the sub-regions will be obtained using a procedure which explicitly minimizes their variance. To accommodate this behavior the miser algorithm allows the total variance to depend on a scaling parameter α,

Var(f) = \frac{σ_{a}}{N_{a}^{α}} + \frac{σ_{b}}{N_{b}^{α}} .

The authors of the original paper describing miser recommend the value α = 2 as a good choice, obtained from numerical experiments, and this is used as the default value in this implementation.

[Variable] double dither

This parameter introduces a random fractional variation of size dither into each bisection, which can be used to break the symmetry of integrands which are concentrated near the exact center of the hypercubic integration region. The default value of dither is zero, so no variation is introduced. If needed, a typical value of dither is 0.1.

23.4 VEGAS

The vegas algorithm of Lepage is based on importance sampling. It samples points from the probability distribution described by the function |f|, so that the points are concentrated in the regions that make the largest contribution to the integral.

In general, if the Monte Carlo integral of f is sampled with points distributed according to a probability distribution described by the function g, we obtain an estimate E_{g}(f; N),

E_{g}(f; N) = E(f/g; N)

with a corresponding variance,

Var_{g}(f; N) = Var(f/g; N).

If the probability distribution is chosen as g = |f|/I(|f|) then it can be shown that the variance V_{g}(f; N) vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.

The vegas algorithm approximates the exact distribution by making a number of passes over the integration region while histogramming the function f. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like K^{d} the probability distribution is approximated by a separable function: g(x_{1}, x_{2}, …) = g_{1}(x_{1})g_{2}(x_{2}) … so that the number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of vegas depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with vegas.

vegas incorporates a number of additional features, and combines both stratified sampling and importance sampling. The integration region is divided into a number of “boxes”, with each box getting a fixed number of points (the goal is 2). Each box can then have a fractional number of bins, but if the ratio of bins-per-box is less than two, Vegas switches to a kind variance reduction (rather than importance sampling).

[Function] gsl_monte_vegas_state * gsl_monte_vegas_alloc (size t dim)

This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.

[Function] int gsl_monte_vegas_init (gsl monte vegas state* s)

This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

[Function] int gsl_monte_vegas_integrate (gsl monte function * f, double *

xl, double * xu, size t dim, size t calls, gsl rng * r, gsl monte vegas state * s, double * result, double * abserr)

This routines uses the vegas Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr. The result and its error estimate are based on a weighted average of independent samples. The chi-squared per degree of freedom for the weighted average is returned via the state struct component, s->chisq, and must be consistent with 1 for the weighted average to be reliable.

[Function] void gsl_monte_vegas_free (gsl monte vegas state * s)

This function frees the memory associated with the integrator state s.

The vegas algorithm computes a number of independent estimates of the integral internally, according to the iterations parameter described below, and returns their weighted average. Random sampling of the integrand can occasionally produce an estimate where the error is zero, particularly if the function is constant in some regions. An estimate with zero error causes the weighted average to break down and must be handled separately. In the original Fortran implementations of vegas the error estimate is made non-zero by substituting a small value (typically 1e-30). The implementation in GSL differs from this and avoids the use of an arbitrary constant—it either assigns the value a weight which is the average weight of the preceding estimates or discards it according to the following procedure,

current estimate has zero error, weighted average has finite error The current estimate is assigned a weight which is the average weight of the preceding estimates.

current estimate has finite error, previous estimates had zero error The previous estimates are discarded and the weighted averaging procedure begins with the current estimate.

current estimate has zero error, previous estimates had zero error The estimates are averaged using the arithmetic mean, but no error is computed.

The vegas algorithm is highly configurable. The following variables can be accessed

through the gsl_monte_vegas_state struct,
[Variable] double result
[Variable] double sigma

These parameters contain the raw value of the integral result and its error sigma from the last iteration of the algorithm.

[Variable] double chisq

This parameter gives the chi-squared per degree of freedom for the weighted estimate of the integral. The value of chisq should be close to 1. A value of chisq which differs significantly from 1 indicates that the values from different iterations are inconsistent. In this case the weighted error will be under-estimated, and further iterations of the algorithm are needed to obtain reliable results.

[Variable] double alpha

The parameter alpha controls the stiffness of the rebinning algorithm. It is typically set between one and two. A value of zero prevents rebinning of the grid. The default value is 1.5.

[Variable] size_t iterations

The number of iterations to perform for each call to the routine. The default value is 5 iterations.

[Variable] int stage

Setting this determines the stage of the calculation. Normally, stage = 0 which begins with a new uniform grid and empty weighted average. Calling vegas with stage = 1 retains the grid from the previous run but discards the weighted average, so that one can “tune” the grid using a relatively small number of points and then do a large run with stage = 1 on the optimized grid. Setting stage = 2 keeps the grid and the weighted average from the previous run, but may increase (or decrease) the number of histogram bins in the grid depending on the number of calls available. Choosing stage = 3 enters at the main loop, so that nothing is changed, and is equivalent to performing additional iterations in a previous call.

[Variable] int mode

The possible choices are GSL_VEGAS_MODE_IMPORTANCE, GSL_VEGAS_MODE_
STRATIFIED, GSL_VEGAS_MODE_IMPORTANCE_ONLY. This determines whether vegas

will use importance sampling or stratified sampling, or whether it can pick on its own. In low dimensions vegas uses strict stratified sampling (more precisely, stratified sampling is chosen if there are fewer than 2 bins per box).

[Variable] int verbose
[Variable] FILE * ostream

These parameters set the level of information printed by vegas. All information is written to the stream ostream. The default setting of verbose is -1, which turns off all output. A verbose value of 0 prints summary information about the weighted average and final result, while a value of 1 also displays the grid coordinates. A value of 2 prints information from the rebinning procedure for each iteration.

Chapter 23: Monte Carlo Integration 273

23.5 Examples

The example program below uses the Monte Carlo routines to estimate the value of the following 3-dimensional integral from the theory of random walks,

I = ∫_{−π}^{+π} \frac{dk_{x}}{2π} ∫_{−π}^{+π} \frac{dk_{y}}{2π} ∫_{−π}^{+π} \frac{dk_{z}}{2π} \frac{1}{(1 − cos(k_{x}) cos(k_{y}) cos(k_{z}))}.

The analytic value of this integral can be shown to be I = Γ(1/4)^{4}/(4π^{3}) = 1.393203929685676859.... The integral gives the mean time spent at the origin by a random walk on a body-centered cubic lattice in three dimensions.

For simplicity we will compute the integral over the region (0, 0, 0) to (π, π, π) and multiply by 8 to obtain the full result. The integral is slowly varying in the middle of the region but has integrable singularities at the corners (0, 0, 0), (0, π, π), (π, 0, π) and (π, π, 0). The Monte Carlo routines only select points which are strictly within the integration region and so no special measures are needed to avoid these singularities.

#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_monte.h>
#include <gsl/gsl_monte_plain.h>
#include <gsl/gsl_monte_miser.h>
#include <gsl/gsl_monte_vegas.h>
/* Computation of the integral,
      I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))
   over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
   is Gamma(1/4)^4/(4 pi^3).  This example is taken from
   C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
   Volume 1", Section 1.1, p21, which cites the original
   paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
   1800 (1977) */
/* For simplicity we compute the integral over the region
   (0,0,0) -> (pi,pi,pi) and multiply by 8 */
double exact = 1.3932039296856768591842462603255;
double
g (double *k, size_t dim, void *params)
{
  double A = 1.0 / (M_PI * M_PI * M_PI);
  return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
}
void
display_results (char *title, double result, double error)
{
  printf ("%s ==================\n", title);
  printf ("result = % .6f\n", result);
  printf ("sigma  = % .6f\n", error);
  printf ("exact  = % .6f\n", exact);
  printf ("error  = % .6f = %.1g sigma\n", result - exact,
          fabs (result - exact) / error);
}

Chapter 23: Monte Carlo Integration 274

int
main (void)
{
double res, err;
double xl[3] = { 0, 0, 0 };
double xu[3] = { M_PI, M_PI, M_PI };
const gsl_rng_type *T;
gsl_rng *r;
gsl_monte_function G = { &g, 3, 0 };
size_t calls = 500000;
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
{
  gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
  gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
                             &res, &err);
  gsl_monte_plain_free (s);
  display_results ("plain", res, err);
}
{
  gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
  gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
                             &res, &err);
  gsl_monte_miser_free (s);
  display_results ("miser", res, err);
}
{
  gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);
  gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
                             &res, &err);
  display_results ("vegas warm-up", res, err);
  printf ("converging...\n");
  do
    {
      gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
                                 &res, &err);
      printf ("result = % .6f sigma = % .6f "
              "chisq/dof = %.1f\n", res, err, s->chisq);
    }
  while (fabs (s->chisq - 1.0) > 0.5);
  display_results ("vegas final", res, err);

    gsl_monte_vegas_free (s);
  }
  return 0;
}

With 500,000 function calls the plain Monte Carlo algorithm achieves a fractional error of 0.6%. The estimated error sigma is consistent with the actual error, and the computed result differs from the true result by about one standard deviation,

plain ==================
result =  1.385867
sigma  =  0.007938
exact  =  1.393204
error  = -0.007337 = 0.9 sigma

The miser algorithm reduces the error by a factor of two, and also correctly estimates the error,

miser ==================
result =  1.390656
sigma  =  0.003743
exact  =  1.393204
error  = -0.002548 = 0.7 sigma

In the case of the vegas algorithm the program uses an initial warm-up run of 10,000 function calls to prepare, or “warm up”, the grid. This is followed by a main run with five iterations of 100,000 function calls. The chi-squared per degree of freedom for the five iterations are checked for consistency with 1, and the run is repeated if the results have not converged. In this case the estimates are consistent on the first pass.

vegas warm-up ==================
result =  1.386925
sigma  =  0.002651
exact  =  1.393204
error  = -0.006278 = 2 sigma
converging...
result =  1.392957 sigma =  0.000452 chisq/dof = 1.1
vegas final ==================
result =  1.392957
sigma  =  0.000452
exact  =  1.393204
error  = -0.000247 = 0.5 sigma

If the value of chisq had differed significantly from 1 it would indicate inconsistent results, with a correspondingly underestimated error. The final estimate from vegas (using a similar number of function calls) is significantly more accurate than the other two algorithms.

23.6 References and Further Reading

The miser algorithm is described in the following article by Press and Farrar, W.H. Press, G.R. Farrar, Recursive Stratified Sampling for Multidimensional Monte Carlo Integration, Computers in Physics, v4 (1990), pp190–195.

The vegas algorithm is described in the following papers, G.P. Lepage, A New Algorithm for Adaptive Multidimensional Integration, Journal of Computational Physics 27, 192–203, (1978) G.P. Lepage, VEGAS: An Adaptive Multi-dimensional Integration Program, Cornell preprint CLNS 80-447, March 1980

24 Simulated Annealing

Stochastic search techniques are used when the structure of a space is not well understood or is not smooth, so that techniques like Newton’s method (which requires calculating Jacobian derivative matrices) cannot be used. In particular, these techniques are frequently used to solve combinatorial optimization problems, such as the traveling salesman problem.

The goal is to find a point in the space at which a real valued energy function (or cost function) is minimized. Simulated annealing is a minimization technique which has given good results in avoiding local minima; it is based on the idea of taking a random walk through the space at successively lower temperatures, where the probability of taking a step is given by a Boltzmann distribution.

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

24.1 Simulated Annealing algorithm

The simulated annealing algorithm takes random walks through the problem space, looking for points with low energies; in these random walks, the probability of taking a step is determined by the Boltzmann distribution,

p = e^{−(E_{i+1}−E_{i})/(kT)}

if E_{i+1} > E_{i}, and p = 1 when E_{i+1} ≤ E_{i}.

In other words, a step will occur if the new energy is lower. If the new energy is higher, the transition can still occur, and its likelihood is proportional to the temperature T and inversely proportional to the energy difference E_{i+1} − E_{i}.

The temperature T is initially set to a high value, and a random walk is carried out at that temperature. Then the temperature is lowered very slightly according to a cooling schedule, for example: T → T/μ_{T} where μ_{T} is slightly greater than 1.

The slight probability of taking a step that gives higher energy is what allows simulated annealing to frequently get out of local minima.

24.2 Simulated Annealing functions

[Function] void gsl_siman_solve (const gsl rng * r, void * x0_p,

gsl siman Efunc t Ef, gsl siman step t take_step, gsl siman metric t

        distance, gsl siman print t print_position, gsl siman copy t copyfunc,

gsl siman copy construct t copy_constructor, gsl siman destroy t

        destructor, size t element_size, gsl siman params t params)

This function performs a simulated annealing search through a given space. The space is specified by providing the functions Ef and distance. The simulated annealing steps are generated using the random number generator r and the function take step. The starting configuration of the system should be given by x0 p. The routine offers two modes for updating configurations, a fixed-size mode and a variable-size mode. In the fixed-size mode the configuration is stored as a single block of memory of size element size. Copies of this configuration are created, copied and destroyed internally using the standard library functions malloc, memcpy and free. The function pointers copyfunc, copy constructor and destructor should be null pointers in fixed-size mode.

In the variable-size mode the functions copyfunc, copy constructor and destructor are used to create, copy and destroy configurations internally. The variable element size should be zero in the variable-size mode. The params structure (described below) controls the run by providing the temperature schedule and other tunable parameters to the algorithm. On exit the best result achieved during the search is placed in *x0_p. If the annealing process has been successful this should be a good approximation to the optimal point in the space. If the function pointer print position is not null, a debugging log will be printed to stdout with the following columns:

     number_of_iterations temperature x x-(*x0_p) Ef(x)

and the output of the function print position itself. If print position is null then no information is printed.

The simulated annealing routines require several user-specified functions to define the configuration space and energy function. The prototypes for these functions are given below.

[Data Type] gsl_siman_Efunc_t

This function type should return the energy of a configuration xp.

     double (*gsl_siman_Efunc_t) (void *xp)
[Data Type] gsl_siman_step_t

This function type should modify the configuration xp using a random step taken from the generator r, up to a maximum distance of step size.

     void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
                               double step_size)
[Data Type] gsl_siman_metric_t

This function type should return the distance between two configurations xp and yp.

     double (*gsl_siman_metric_t) (void *xp, void *yp)
[Data Type] gsl_siman_print_t

This function type should print the contents of the configuration xp.

     void (*gsl_siman_print_t) (void *xp)
[Data Type] gsl_siman_copy_t

This function type should copy the configuration source into dest.

     void (*gsl_siman_copy_t) (void *source, void *dest)
[Data Type] gsl_siman_copy_construct_t

This function type should create a new copy of the configuration xp.

     void * (*gsl_siman_copy_construct_t) (void *xp)
[Data Type] gsl_siman_destroy_t

This function type should destroy the configuration xp, freeing its memory.

     void (*gsl_siman_destroy_t) (void *xp)

[Data Type] gsl_siman_params_t

These are the parameters that control a run of gsl_siman_solve. This structure contains all the information needed to control the search, beyond the energy function, the step function and the initial guess.

     int n_tries

The number of points to try for each step.

     int iters_fixed_T

The number of iterations at each temperature.

     double step_size

The maximum step size in the random walk.

     double k, t_initial, mu_t, t_min

The parameters of the Boltzmann distribution and cooling schedule.

24.3 Examples

The simulated annealing package is clumsy, and it has to be because it is written in C, for C callers, and tries to be polymorphic at the same time. But here we provide some examples which can be pasted into your application with little change and should make things easier.

24.3.1 Trivial example

The first example, in one dimensional cartesian space, sets up an energy function which is a damped sine wave; this has many local minima, but only one global minimum, somewhere between 1.0 and 1.5. The initial guess given is 15.5, which is several local minima away from the global minimum.

      #include <math.h>
      #include <stdlib.h>
      #include <gsl/gsl_siman.h>
      /* set up parameters for this simulated annealing run */
      /* how many points do we try before stepping */
      #define N_TRIES 200
      /* how many iterations for each T? */
      #define ITERS_FIXED_T 10
      /* max step size in random walk */
      #define STEP_SIZE 10
      /* Boltzmann constant */
      #define K 1.0
      /* initial temperature */
      #define T_INITIAL 0.002
      /* damping factor for temperature */
      #define MU_T 1.005
      #define T_MIN 2.0e-6
      gsl_siman_params_t params
        = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,

Chapter 24: Simulated Annealing 280

   K, T_INITIAL, MU_T, T_MIN};
/* now some functions to test in one dimension */
double E1(void *xp)
{
double x = * ((double *) xp);
return exp(-pow((x-1.0),2.0))*sin(8*x);
}
double M1(void *xp, void *yp)
{
double x = *((double *) xp);
double y = *((double *) yp);
return fabs(x - y);
}
void S1(const gsl_rng * r, void *xp, double step_size)
{
double old_x = *((double *) xp);
double new_x;
double u = gsl_rng_uniform(r);
new_x = u * 2 * step_size - step_size + old_x;
memcpy(xp, &new_x, sizeof(new_x));
}
void P1(void *xp)
{
printf ("%12g", *((double *) xp));
}
int
main(int argc, char *argv[])
{
const gsl_rng_type * T;
gsl_rng * r;
double x_initial = 15.5;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc(T);
gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
                NULL, NULL, NULL,
                sizeof(double), params);
return 0;
}

Here are a couple of plots that are generated by running siman_test in the following way:

     $ ./siman_test | grep -v "^#"
      | xyplot -xyil -y -0.88 -0.83 -d "x...y"
      | xyps -d > siman-test.eps
     $ ./siman_test | grep -v "^#"
      | xyplot -xyil -xl "generation" -yl "energy" -d "x..y"
      | xyps -d > siman-energy.eps

Example of a simulated annealing run: at higher temperatures (early in the plot) you see that the solution can fluctuate, but at lower temperatures it converges.

24.3.2 Traveling Salesman Problem

The TSP (Traveling Salesman Problem) is the classic combinatorial optimization problem. I have provided a very simple version of it, based on the coordinates of twelve cities in the southwestern United States. This should maybe be called the Flying Salesman Problem, Chapter 24: Simulated Annealing 282

since I am using the great-circle distance between cities, rather than the driving distance. Also: I assume the earth is a sphere, so I don’t use geoid distances.

The gsl_siman_solve() routine finds a route which is 3490.62 Kilometers long; this is confirmed by an exhaustive search of all possible routes with the same initial city.

The full code can be found in ‘siman/siman_tsp.c’, but I include here some plots generated in the following way:

$ ./siman_tsp > tsp.output
$ grep -v "^#" tsp.output
 | xyplot -xyil -d "x................y"
    -lx "generation" -ly "distance"
    -lt "TSP -- 12 southwest cities"
 | xyps -d > 12-cities.eps
$ grep initial_city_coord tsp.output
 | awk '{print $2, $3, $4, $5}'
 | xyplot -xyil -lb0 -cs 0.8
    -lx "longitude (- means west)" -ly "latitude"
    -lt "TSP -- initial-order"
 | xyps -d > initial-route.eps
$ grep final_city_coord tsp.output
 | awk '{print $2, $3, $4, $5}'
 | xyplot -xyil -lb0 -cs 0.8
    -lx "longitude (- means west)" -ly "latitude"
    -lt "TSP -- final-order"
 | xyps -d > final-route.eps

This is the output showing the initial order of the cities; longitude is negative, since it is west and I want the plot to look like a map.

# initial coordinates of cities (longitude and latitude)
###initial_city_coord: -105.95 35.68 Santa Fe
###initial_city_coord: -112.07 33.54 Phoenix
###initial_city_coord: -106.62 35.12 Albuquerque
###initial_city_coord: -103.2 34.41 Clovis
###initial_city_coord: -107.87 37.29 Durango
###initial_city_coord: -96.77 32.79 Dallas
###initial_city_coord: -105.92 35.77 Tesuque
###initial_city_coord: -107.84 35.15 Grants
###initial_city_coord: -106.28 35.89 Los Alamos
###initial_city_coord: -106.76 32.34 Las Cruces
###initial_city_coord: -108.58 37.35 Cortez
###initial_city_coord: -108.74 35.52 Gallup
###initial_city_coord: -105.95 35.68 Santa Fe

The optimal route turns out to be:

# final coordinates of cities (longitude and latitude)
###final_city_coord: -105.95 35.68 Santa Fe
###final_city_coord: -106.28 35.89 Los Alamos
###final_city_coord: -106.62 35.12 Albuquerque
###final_city_coord: -107.84 35.15 Grants
###final_city_coord: -107.87 37.29 Durango
###final_city_coord: -108.58 37.35 Cortez
###final_city_coord: -108.74 35.52 Gallup
###final_city_coord: -112.07 33.54 Phoenix
###final_city_coord: -106.76 32.34 Las Cruces
###final_city_coord: -96.77 32.79 Dallas
###final_city_coord: -103.2 34.41 Clovis
###final_city_coord: -105.92 35.77 Tesuque
###final_city_coord: -105.95 35.68 Santa Fe

Initial and final (optimal) route for the 12 southwestern cities Flying Salesman Problem.

Here’s a plot of the cost function (energy) versus generation (point in the calculation at which a new temperature is set) for this problem:

Example of a simulated annealing run for the 12 southwestern cities Flying Salesman Problem.

24.4 References and Further Reading

Further information is available in the following book, Chapter 24: Simulated Annealing 284

Modern Heuristic Techniques for Combinatorial Problems, Colin R. Reeves (ed.), McGraw-Hill, 1995 (ISBN 0-07-709239-2).

25 Ordinary Differential Equations

This chapter describes functions for solving ordinary differential equation (ODE) initial value problems. The library provides a variety of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines, and higher-level components for adaptive step-size control. The components can be combined by the user to achieve the desired solution, with full access to any intermediate steps.

These functions are declared in the header file ‘gsl_odeiv.h’.

25.1 Defining the ODE System

The routines solve the general n-dimensional first-order system,

\frac{dy_{i}(t)}{dt} = f_{i}(t, y_{1}(t), … y_{n}(t))

for i = 1, …, n. The stepping functions rely on the vector of derivatives f_{i} and the Jacobian matrix, J_{ij} = ∂f_{i}(t, y(t))/∂y_{j}. A system of equations is defined using the gsl_odeiv_system datatype.

[Data Type] gsl_odeiv_system

This data type defines a general ODE system with arbitrary parameters.

int (* function) (double t, const double y[], double dydt[], void *
params)

This function should store the vector elements f_{i}(t, y, params) in the array dydt, for arguments (t,y) and parameters params. The function should return GSL_SUCCESS if the calculation was completed successfully. Any other return value indicates an error.

int (* jacobian) (double t, const double y[], double * dfdy, double
dfdt[], void * params);

This function should store the vector of derivative elements ∂f_{i}(t, y, params)/∂t in the array dfdt and the Jacobian matrix J_{ij} in the array dfdy, regarded as a row-ordered matrix J(i,j) = dfdy[i * dimension + j] where dimension is the dimension of the system. The function should return GSL_SUCCESS if the calculation was completed successfully. Any other return value indicates an error. Some of the simpler solver algorithms do not make use of the Jacobian matrix, so it is not always strictly necessary to provide it (the jacobian element of the struct can be replaced by a null pointer for those algorithms). However, it is useful to provide the Jacobian to allow the solver algorithms to be interchanged—the best algorithms make use of the Jacobian.

size_t dimension;

This is the dimension of the system of equations.

void * params

This is a pointer to the arbitrary parameters of the system.

25.2 Stepping Functions

The lowest level components are the stepping functions which advance a solution from time t to t + h for a fixed step-size h and estimate the resulting local error.

[Function] gsl_odeiv_step * gsl_odeiv_step_alloc (const gsl odeiv step type

* T, size t dim) This function returns a pointer to a newly allocated instance of a stepping function of type T for a system of dim dimensions.

[Function] int gsl_odeiv_step_reset (gsl odeiv step * s)

This function resets the stepping function s. It should be used whenever the next use of s will not be a continuation of a previous step.

[Function] void gsl_odeiv_step_free (gsl odeiv step * s)

This function frees all the memory associated with the stepping function s.

[Function] const char * gsl_odeiv_step_name (const gsl odeiv step * s)

This function returns a pointer to the name of the stepping function. For example,

     printf ("step method is '%s'\n",
              gsl_odeiv_step_name (s));

would print something like step method is 'rk4'.

[Function] unsigned int gsl_odeiv_step_order (const gsl odeiv step * s)

This function returns the order of the stepping function on the previous step. This order can vary if the stepping function itself is adaptive.

[Function] int gsl_odeiv_step_apply (gsl odeiv step * s, double t, double h, double y[], double yerr[], const double dydt_in[], double dydt_out[], const gsl odeiv system * dydt)

This function applies the stepping function s to the system of equations defined by dydt, using the step size h to advance the system from time t and state y to time t+h. The new state of the system is stored in y on output, with an estimate of the absolute error in each component stored in yerr. If the argument dydt in is not null it should point an array containing the derivatives for the system at time t on input. This is optional as the derivatives will be computed internally if they are not provided, but allows the reuse of existing derivative information. On output the new derivatives of the system at time t+h will be stored in dydt out if it is not null. If the user-supplied functions defined in the system dydt return a status other than GSL_SUCCESS the step will be aborted. In this case, the elements of y will be restored to their pre-step values and the error code from the user-supplied function will be returned. To distinguish between error codes from the user-supplied functions and those from gsl_odeiv_step_apply itself, any user-defined return values should be distinct from the standard GSL error codes.

The following algorithms are available,

[Step Type] gsl_odeiv_step_rk2

Embedded Runge-Kutta (2, 3) method.

[Step Type] gsl_odeiv_step_rk4

4th order (classical) Runge-Kutta.

[Step Type] gsl_odeiv_step_rkf45

Embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good generalpurpose integrator.

[Step Type] gsl_odeiv_step_rkck

Embedded Runge-Kutta Cash-Karp (4, 5) method.

[Step Type] gsl_odeiv_step_rk8pd

Embedded Runge-Kutta Prince-Dormand (8,9) method.

[Step Type] gsl_odeiv_step_rk2imp

Implicit 2nd order Runge-Kutta at Gaussian points.

[Step Type] gsl_odeiv_step_rk4imp

Implicit 4th order Runge-Kutta at Gaussian points.

[Step Type] gsl_odeiv_step_bsimp

Implicit Bulirsch-Stoer method of Bader and Deuflhard. This algorithm requires the Jacobian.

[Step Type] gsl_odeiv_step_gear1

M=1 implicit Gear method.

[Step Type] gsl_odeiv_step_gear2

M=2 implicit Gear method.

25.3 Adaptive Step-size Control

The control function examines the proposed change to the solution produced by a stepping function and attempts to determine the optimal step-size for a user-specified level of error.

[Function] gsl_odeiv_control * gsl_odeiv_control_standard_new (double
   eps_abs, double eps_rel, double a_y, double a_dydt)

The standard control object is a four parameter heuristic based on absolute and relative errors eps abs and eps rel, and scaling factors a y and a dydt for the system state y(t) and derivatives y′(t) respectively. The step-size adjustment procedure for this method begins by computing the desired error level D_{i} for each component,

D_{i} = ϵ_{abs} + ϵ_{rel} ∗ (a_{y}|y_{i}| + a_{dydt}h|y′_{i}|)

and comparing it with the observed error E_{i} = |yerr_{i}|. If the observed error E exceeds the desired error level D by more than 10% for any component then the method reduces the step-size by an appropriate factor,

h_{new} = h_{old} ∗ S ∗ (E/D)^{−1/q}

where q is the consistency order of the method (e.g. q = 4 for 4(5) embedded RK), and S is a safety factor of 0.9. The ratio E/D is taken to be the maximum of the ratios E_{i}/D_{i}.

If the observed error E is less than 50% of the desired error level D for the maximum ratio E_{i}/D_{i} then the algorithm takes the opportunity to increase the step-size to bring the error in line with the desired level,

h_{new} = h_{old} ∗ S ∗ (E/D)^{−1/(q+1)}

This encompasses all the standard error scaling methods. To avoid uncontrolled changes in the stepsize, the overall scaling factor is limited to the range 1/5 to 5.

[Function] gsl_odeiv_control * gsl_odeiv_control_y_new (double eps_abs,
   double eps_rel)

This function creates a new control object which will keep the local error on each step within an absolute error of eps abs and relative error of eps rel with respect to the solution y_{i}(t). This is equivalent to the standard control object with a y=1 and a dydt=0.

[Function] gsl_odeiv_control * gsl_odeiv_control_yp_new (double
   eps_abs, double eps_rel)

This function creates a new control object which will keep the local error on each step within an absolute error of eps abs and relative error of eps rel with respect to the derivatives of the solution y′_{i}(t). This is equivalent to the standard control object with a y=0 and a dydt=1.

[Function] gsl_odeiv_control * gsl_odeiv_control_scaled_new (double

eps_abs, double eps_rel, double a_y, double a_dydt, const double

   scale_abs[], size t dim)

This function creates a new control object which uses the same algorithm as gsl_ odeiv_control_standard_new but with an absolute error which is scaled for each component by the array scale abs. The formula for D_{i} for this control object is,

D_{i} = ϵ_{abs}s_{i} + ϵ_{rel} ∗ (a_{y}|y_{i}| + a_{dydt}h|y′_{i}|)

where s_{i} is the i-th component of the array scale abs. The same error control heuristic is used by the Matlab ode suite.

[Function] gsl_odeiv_control * gsl_odeiv_control_alloc (const

gsl odeiv control type * T) This function returns a pointer to a newly allocated instance of a control function of type T. This function is only needed for defining new types of control functions. For most purposes the standard control functions described above should be sufficient.

[Function] int gsl_odeiv_control_init (gsl odeiv control * c, double eps_abs,

double eps_rel, double a_y, double a_dydt) This function initializes the control function c with the parameters eps abs (absolute error), eps rel (relative error), a y (scaling factor for y) and a dydt (scaling factor for derivatives).

[Function] void gsl_odeiv_control_free (gsl odeiv control * c)

This function frees all the memory associated with the control function c.

[Function] int gsl_odeiv_control_hadjust (gsl odeiv control * c, gsl odeiv step

* s, const double y0[], const double yerr[], const double dydt[], double * h) This function adjusts the step-size h using the control function c, and the current values of y, yerr and dydt. The stepping function step is also needed to determine the order of the method. If the error in the y-values yerr is found to be too large then the step-size h is reduced and the function returns GSL_ODEIV_HADJ_DEC. If the error is sufficiently small then h may be increased and GSL_ODEIV_HADJ_INC is returned. The function returns GSL_ODEIV_HADJ_NIL if the step-size is unchanged. The goal of the function is to estimate the largest step-size which satisfies the user-specified accuracy requirements for the current point.

[Function] const char * gsl_odeiv_control_name (const gsl odeiv control * c)

This function returns a pointer to the name of the control function. For example,

     printf ("control method is '%s'\n",
             gsl_odeiv_control_name (c));
would print something like control method is 'standard'

25.4 Evolution

The highest level of the system is the evolution function which combines the results of a stepping function and control function to reliably advance the solution forward over an interval (t_{0}, t_{1}). If the control function signals that the step-size should be decreased the evolution function backs out of the current step and tries the proposed smaller step-size. This process is continued until an acceptable step-size is found.

[Function] gsl_odeiv_evolve * gsl_odeiv_evolve_alloc (size t dim)

This function returns a pointer to a newly allocated instance of an evolution function for a system of dim dimensions.

[Function] int gsl_odeiv_evolve_apply (gsl odeiv evolve * e, gsl odeiv control * con, gsl odeiv step * step, const gsl odeiv system * dydt, double * t, double t1, double * h, double y[])

This function advances the system (e, dydt) from time t and position y using the stepping function step. The new time and position are stored in t and y on output. The initial step-size is taken as h, but this will be modified using the control function c to achieve the appropriate error bound if necessary. The routine may make several calls to step in order to determine the optimum step-size. If the step-size has been changed the value of h will be modified on output. The maximum time t1 is guaranteed not to be exceeded by the time-step. On the final time-step the value of t will be set to t1 exactly.

If the user-supplied functions defined in the system dydt return a status other than GSL_SUCCESS the step will be aborted. In this case, t and y will be restored to their pre-step values and the error code from the user-supplied function will be returned. To distinguish between error codes from the user-supplied functions and those from gsl_odeiv_evolve_apply itself, any user-defined return values should be distinct from the standard GSL error codes.

[Function] int gsl_odeiv_evolve_reset (gsl odeiv evolve * e)

This function resets the evolution function e. It should be used whenever the next use of e will not be a continuation of a previous step.

[Function] void gsl_odeiv_evolve_free (gsl odeiv evolve * e)

This function frees all the memory associated with the evolution function e.

25.5 Examples

The following program solves the second-order nonlinear Van der Pol oscillator equation,

x′′(t) + μx′(t)(x(t)^{2} − 1) + x(t) = 0

This can be converted into a first order system suitable for use with the routines described in this chapter by introducing a separate variable for the velocity, y = x′(t),

\begin{aligned}x′ & = y \\ y′ & = −x + μy(1 − x^{2}) \end{aligned}

The program begins by defining functions for these derivatives and their Jacobian,

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv.h>
int
func (double t, const double y[], double f[],
      void *params)
{
  double mu = *(double *)params;
  f[0] = y[1];
  f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
  return GSL_SUCCESS;
}
int
jac (double t, const double y[], double *dfdy,
     double dfdt[], void *params)
{
  double mu = *(double *)params;
  gsl_matrix_view dfdy_mat
    = gsl_matrix_view_array (dfdy, 2, 2);
  gsl_matrix * m = &dfdy_mat.matrix;
  gsl_matrix_set (m, 0, 0, 0.0);
  gsl_matrix_set (m, 0, 1, 1.0);
  gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
  gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
  dfdt[0] = 0.0;
  dfdt[1] = 0.0;

return GSL_SUCCESS;
}
int
main (void)
{
const gsl_odeiv_step_type * T
  = gsl_odeiv_step_rk8pd;
gsl_odeiv_step * s
  = gsl_odeiv_step_alloc (T, 2);
gsl_odeiv_control * c
  = gsl_odeiv_control_y_new (1e-6, 0.0);
gsl_odeiv_evolve * e
  = gsl_odeiv_evolve_alloc (2);
double mu = 10;
gsl_odeiv_system sys = {func, jac, 2, &mu};
double t = 0.0, t1 = 100.0;
double h = 1e-6;
double y[2] = { 1.0, 0.0 };
while (t < t1)
  {
    int status = gsl_odeiv_evolve_apply (e, c, s,
                                         &sys,
                                         &t, t1,
                                         &h, y);
    if (status != GSL_SUCCESS)
        break;
    printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
  }
gsl_odeiv_evolve_free (e);
gsl_odeiv_control_free (c);
gsl_odeiv_step_free (s);
return 0;
}

For functions with multiple parameters, the appropriate information can be passed in through the params argument using a pointer to a struct.

The main loop of the program evolves the solution from (y, y′) = (1, 0) at t = 0 to t = 100. The step-size h is automatically adjusted by the controller to maintain an absolute accuracy of 10^{−6} in the function values y.

Numerical solution of the Van der Pol oscillator equation using Prince-Dormand 8th order Runge-Kutta.

To obtain the values at regular intervals, rather than the variable spacings chosen by the control function, the main loop can be modified to advance the solution from one point to the next. For example, the following main loop prints the solution at the fixed points t = 0, 1, 2, …, 100,

       for (i = 1; i <= 100; i++)
         {
           double ti = i * t1 / 100.0;
           while (t < ti)
             {
               gsl_odeiv_evolve_apply (e, c, s,
                                       &sys,
                                       &t, ti, &h,
                                       y);
             }
           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

It is also possible to work with a non-adaptive integrator, using only the stepping function itself. The following program uses the rk4 fourth-order Runge-Kutta stepping function with a fixed stepsize of 0.01,

     int
     main (void)
     {
       const gsl_odeiv_step_type * T
         = gsl_odeiv_step_rk4;
       gsl_odeiv_step * s
         = gsl_odeiv_step_alloc (T, 2);
       double mu = 10;
       gsl_odeiv_system sys = {func, jac, 2, &mu};

double t = 0.0, t1 = 100.0;
double h = 1e-2;
double y[2] = { 1.0, 0.0 }, y_err[2];
double dydt_in[2], dydt_out[2];
/* initialise dydt_in from system parameters */
GSL_ODEIV_FN_EVAL(&sys, t, y, dydt_in);
while (t < t1)
  {
    int status = gsl_odeiv_step_apply (s, t, h,
                                       y, y_err,
                                       dydt_in,
                                       dydt_out,
                                       &sys);
    if (status != GSL_SUCCESS)
        break;
    dydt_in[0] = dydt_out[0];
    dydt_in[1] = dydt_out[1];
    t += h;
    printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
  }
gsl_odeiv_step_free (s);
return 0;
}

The derivatives must be initialized for the starting point t = 0 before the first step is taken. Subsequent steps use the output derivatives dydt out as inputs to the next step by copying their values into dydt in.

25.6 References and Further Reading

Many of the basic Runge-Kutta formulas can be found in the Handbook of Mathematical Functions,

Abramowitz & Stegun (eds.), Handbook of Mathematical Functions, Section 25.5.

The implicit Bulirsch-Stoer algorithm bsimp is described in the following paper, G. Bader and P. Deuflhard, “A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations.”, Numer. Math. 41, 373–398, 1983.

26 Interpolation

This chapter describes functions for performing interpolation. The library provides a variety of interpolation methods, including Cubic splines and Akima splines. The interpolation types are interchangeable, allowing different methods to be used without recompiling. Interpolations can be defined for both normal and periodic boundary conditions. Additional functions are available for computing derivatives and integrals of interpolating functions.

The functions described in this section are declared in the header files ‘gsl_interp.h’

and ‘gsl_spline.h’.

26.1 Introduction

Given a set of data points (x_{1}, y_{1}) … (x_{n}, y_{n}) the routines described in this section compute a continuous interpolating function y(x) such that y(x_{i}) = y_{i}. The interpolation is piecewise smooth, and its behavior at the end-points is determined by the type of interpolation used.

26.2 Interpolation Functions

The interpolation function for a given dataset is stored in a gsl_interp object. These are created by the following functions.

[Function] gsl_interp * gsl_interp_alloc (const gsl interp type * T, size t

        size)

This function returns a pointer to a newly allocated interpolation object of type T for size data-points.

[Function] int gsl_interp_init (gsl interp * interp, const double xa[], const

double ya[], size t size) This function initializes the interpolation object interp for the data (xa,ya) where xa and ya are arrays of size size. The interpolation object (gsl_interp) does not save the data arrays xa and ya and only stores the static state computed from the data. The xa data array is always assumed to be strictly ordered; the behavior for other arrangements is not defined.

[Function] void gsl_interp_free (gsl interp * interp)

This function frees the interpolation object interp.

26.3 Interpolation Types

The interpolation library provides five interpolation types:

[Interpolation Type] gsl_interp_linear

Linear interpolation. This interpolation method does not require any additional memory.

[Interpolation Type] gsl_interp_polynomial

Polynomial interpolation. This method should only be used for interpolating small numbers of points because polynomial interpolation introduces large oscillations, even for well-behaved datasets. The number of terms in the interpolating polynomial is equal to the number of points.

[Interpolation Type] gsl_interp_cspline

Cubic spline with natural boundary conditions. The resulting curve is piecewise cubic on each interval, with matching first and second derivatives at the supplied data-points. The second derivative is chosen to be zero at the first point and last point.

[Interpolation Type] gsl_interp_cspline_periodic

Cubic spline with periodic boundary conditions. The resulting curve is piecewise cubic on each interval, with matching first and second derivatives at the supplied data-points. The derivatives at the first and last points are also matched. Note that the last point in the data must have the same y-value as the first point, otherwise the resulting periodic interpolation will have a discontinuity at the boundary.

[Interpolation Type] gsl_interp_akima

Non-rounded Akima spline with natural boundary conditions. This method uses the non-rounded corner algorithm of Wodicka.

[Interpolation Type] gsl_interp_akima_periodic

Non-rounded Akima spline with periodic boundary conditions. This method uses the non-rounded corner algorithm of Wodicka.

The following related functions are available:

[Function] const char * gsl_interp_name (const gsl interp * interp)

This function returns the name of the interpolation type used by interp. For example,

     printf ("interp uses '%s' interpolation.\n",
             gsl_interp_name (interp));

would print something like,

     interp uses 'cspline' interpolation.
[Function] unsigned int gsl_interp_min_size (const gsl interp * interp)

This function returns the minimum number of points required by the interpolation type of interp. For example, Akima spline interpolation requires a minimum of 5 points.

26.4 Index Look-up and Acceleration

The state of searches can be stored in a gsl_interp_accel object, which is a kind of iterator for interpolation lookups. It caches the previous value of an index lookup. When the subsequent interpolation point falls in the same interval its index value can be returned immediately.

[Function] size_t gsl_interp_bsearch (const double x_array[], double x, size t

   index_lo, size t index_hi)

This function returns the index i of the array x array such that x_array[i] <= x < x_array[i+1]. The index is searched for in the range [index lo,index hi].

[Function] gsl_interp_accel * gsl_interp_accel_alloc (void)

This function returns a pointer to an accelerator object, which is a kind of iterator for interpolation lookups. It tracks the state of lookups, thus allowing for application of various acceleration strategies.

[Function] size_t gsl_interp_accel_find (gsl interp accel * a, const double

x_array[], size t size, double x)

This function performs a lookup action on the data array x array of size size, using the given accelerator a. This is how lookups are performed during evaluation of an interpolation. The function returns an index i such that x_array[i] <= x < x_

array[i+1].
[Function] void gsl_interp_accel_free (gsl interp accel* acc)

This function frees the accelerator object acc.

26.5 Evaluation of Interpolating Functions

[Function] double gsl_interp_eval (const gsl interp * interp, const double xa[],

const double ya[], double x, gsl interp accel * acc)

[Function] int gsl_interp_eval_e (const gsl interp * interp, const double xa[],

const double ya[], double x, gsl interp accel * acc, double * y) These functions return the interpolated value of y for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator acc.

[Function] double gsl_interp_eval_deriv (const gsl interp * interp, const

double xa[], const double ya[], double x, gsl interp accel * acc)

[Function] int gsl_interp_eval_deriv_e (const gsl interp * interp, const

double xa[], const double ya[], double x, gsl interp accel * acc, double * d) These functions return the derivative d of an interpolated function for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator acc.

[Function] double gsl_interp_eval_deriv2 (const gsl interp * interp, const

double xa[], const double ya[], double x, gsl interp accel * acc)

[Function] int gsl_interp_eval_deriv2_e (const gsl interp * interp, const

double xa[], const double ya[], double x, gsl interp accel * acc, double * d2) These functions return the second derivative d2 of an interpolated function for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator acc.

[Function] double gsl_interp_eval_integ (const gsl interp * interp, const

double xa[], const double ya[], double a, double b, gsl interp accel * acc)

[Function] int gsl_interp_eval_integ_e (const gsl interp * interp, const

double xa[], const double ya[], double a, double b, gsl interp accel * acc, double * result)

These functions return the numerical integral result of an interpolated function over the range [a, b], using the interpolation object interp, data arrays xa and ya and the accelerator acc.

26.6 Higher-level Interface

The functions described in the previous sections required the user to supply pointers to the x and y arrays on each call. The following functions are equivalent to the corresponding gsl_interp functions but maintain a copy of this data in the gsl_spline object. This removes the need to pass both xa and ya as arguments on each evaluation. These functions are defined in the header file ‘gsl_spline.h’.

[Function] gsl_spline * gsl_spline_alloc (const gsl interp type * T, size t

        size)

[Function] int gsl_spline_init (gsl spline * spline, const double xa[], const double ya[], size t size)

[Function] void gsl_spline_free (gsl spline * spline)
[Function] const char * gsl_spline_name (const gsl spline * spline)
[Function] unsigned int gsl_spline_min_size (const gsl spline * spline)

[Function] double gsl_spline_eval (const gsl spline * spline, double x,

gsl interp accel * acc)

[Function] int gsl_spline_eval_e (const gsl spline * spline, double x,

gsl interp accel * acc, double * y)

[Function] double gsl_spline_eval_deriv (const gsl spline * spline, double x,

gsl interp accel * acc)

[Function] int gsl_spline_eval_deriv_e (const gsl spline * spline, double x,

gsl interp accel * acc, double * d)

[Function] double gsl_spline_eval_deriv2 (const gsl spline * spline, double x,

gsl interp accel * acc)

[Function] int gsl_spline_eval_deriv2_e (const gsl spline * spline, double x,

gsl interp accel * acc, double * d2)

[Function] double gsl_spline_eval_integ (const gsl spline * spline, double a,

double b, gsl interp accel * acc)

[Function] int gsl_spline_eval_integ_e (const gsl spline * spline, double a,

double b, gsl interp accel * acc, double * result)

26.7 Examples

The following program demonstrates the use of the interpolation and spline functions. It computes a cubic spline interpolation of the 10-point dataset (x_{i}, y_{i}) where x_{i} = i + sin(i)/2 and y_{i} = i + cos(i^{2}) for i = 0 … 9.

     #include <stdlib.h>
     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_spline.h>
     int
     main (void)
     {
       int i;
       double xi, yi, x[10], y[10];

printf ("#m=0,S=2\n");
for (i = 0; i < 10; i++)
{
  x[i] = i + 0.5 * sin (i);
  y[i] = i + cos (i * i);
  printf ("%g %g\n", x[i], y[i]);
}
printf ("#m=1,S=0\n");
{
gsl_interp_accel *acc
  = gsl_interp_accel_alloc ();
gsl_spline *spline
  = gsl_spline_alloc (gsl_interp_cspline, 10);
gsl_spline_init (spline, x, y, 10);
for (xi = x[0]; xi < x[9]; xi += 0.01)
  {
    yi = gsl_spline_eval (spline, xi, acc);
    printf ("%g %g\n", xi, yi);
  }
gsl_spline_free (spline);
gsl_interp_accel_free (acc);
}
return 0;
}

The output is designed to be used with the gnu plotutils graph program,

$ ./a.out > interp.dat
$ graph -T ps < interp.dat > interp.ps

The result shows a smooth interpolation of the original points. The interpolation method can changed simply by varying the first argument of gsl_spline_alloc.

The next program demonstrates a periodic cubic spline with 4 data points. Note that the first and last points must be supplied with the same y-value for a periodic spline.

#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_spline.h>
int
main (void)
{
int N = 4;
double x[4] = {0.00, 0.10,  0.27,  0.30};
double y[4] = {0.15, 0.70, -0.10,  0.15}; /* Note: first = last
                                             for periodic data */
gsl_interp_accel *acc = gsl_interp_accel_alloc ();
const gsl_interp_type *t = gsl_interp_cspline_periodic;
gsl_spline *spline = gsl_spline_alloc (t, N);
int i; double xi, yi;
printf ("#m=0,S=5\n");
for (i = 0; i < N; i++)
  {
    printf ("%g %g\n", x[i], y[i]);
  }
printf ("#m=1,S=0\n");
gsl_spline_init (spline, x, y, N);
for (i = 0; i <= 100; i++)
  {
    xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
    yi = gsl_spline_eval (spline, xi, acc);
    printf ("%g %g\n", xi, yi);
  }
gsl_spline_free (spline);
gsl_interp_accel_free (acc);
return 0;
}

The output can be plotted with gnu graph.

$ ./a.out > interp.dat
$ graph -T ps < interp.dat > interp.ps

The result shows a periodic interpolation of the original points. The slope of the fitted curve is the same at the beginning and end of the data, and the second derivative is also.

26.8 References and Further Reading

Descriptions of the interpolation algorithms and further references can be found in the following books:

C.W. Ueberhuber, Numerical Computation (Volume 1), Chapter 9 “Interpolation”, Springer (1997), ISBN 3-540-62058-3. D.M. Young, R.T. Gregory A Survey of Numerical Mathematics (Volume 1), Chapter 6.8, Dover (1988), ISBN 0-486-65691-8.

27 Numerical Differentiation

The functions described in this chapter compute numerical derivatives by finite differencing. An adaptive algorithm is used to find the best choice of finite difference and to estimate the error in the derivative. These functions are declared in the header file ‘gsl_deriv.h’.

27.1 Functions

[Function] int gsl_deriv_central (const gsl function * f, double x, double h,

double * result, double * abserr) This function computes the numerical derivative of the function f at the point x using an adaptive central difference algorithm with a step-size of h. The derivative is returned in result and an estimate of its absolute error is returned in abserr.

The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative is computed using a 5-point rule for equally spaced abscissae at x − h, x − h/2, x, x + h/2, x, with an error estimate taken from the difference between the 5-point rule and the corresponding 3-point rule x−h, x, x+h. Note that the value of the function at x does not contribute to the derivative calculation, so only 4-points are actually used.

[Function] int gsl_deriv_forward (const gsl function * f, double x, double h,

double * result, double * abserr) This function computes the numerical derivative of the function f at the point x using an adaptive forward difference algorithm with a step-size of h. The function is evaluated only at points greater than x, and never at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a discontinuity at x, or is undefined for values less than x.

The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative at x is computed using an “open” 4-point rule for equally spaced abscissae at x+h/4, x + h/2, x + 3h/4, x + h, with an error estimate taken from the difference between the 4-point rule and the corresponding 2-point rule x + h/2, x + h.

[Function] int gsl_deriv_backward (const gsl function * f, double x, double h,

double * result, double * abserr) This function computes the numerical derivative of the function f at the point x using an adaptive backward difference algorithm with a step-size of h. The function is evaluated only at points less than x, and never at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a discontinuity at x, or is undefined for values greater than x.

This function is equivalent to calling gsl_deriv_forward with a negative step-size.

27.2 Examples

The following code estimates the derivative of the function f(x) = x^{3/2} at x = 2 and at x = 0. The function f(x) is undefined for x < 0 so the derivative at x = 0 is computed

using gsl_deriv_forward.
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_deriv.h>
double f (double x, void * params)
{
  return pow (x, 1.5);
}
int
main (void)
{
  gsl_function F;
  double result, abserr;
  F.function = &f;
  F.params = 0;
  printf ("f(x) = x^(3/2)\n");
  gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
  printf ("x = 2.0\n");
  printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
  printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));
  gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
  printf ("x = 0.0\n");
  printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
  printf ("exact = %.10f\n", 0.0);
  return 0;
}

Here is the output of the program,

$ ./a.out
f(x) = x^(3/2)
x = 2.0
f'(x) = 2.1213203120 +/- 0.0000004064
exact = 2.1213203436
x = 0.0
f'(x) = 0.0000000160 +/- 0.0000000339
exact = 0.0000000000

27.3 References and Further Reading

The algorithms used by these functions are described in the following sources: Abramowitz and Stegun, Handbook of Mathematical Functions, Section 25.3.4, and Table 25.5 (Coefficients for Differentiation). S.D. Conte and Carl de Boor, Elementary Numerical Analysis: An Algorithmic Approach, McGraw-Hill, 1972.

28 Chebyshev Approximations

This chapter describes routines for computing Chebyshev approximations to univariate functions. A Chebyshev approximation is a truncation of the series f(x) = ∑ c_{n}T_{n}(x), where the Chebyshev polynomials T_{n}(x) = cos(n arccos x) provide an orthogonal basis of polynomials on the interval [−1, 1] with the weight function 1/\sqrt{1 − x^{2}}. The first few Chebyshev polynomials are, T_{0}(x) = 1, T_{1}(x) = x, T_{2}(x) = 2x^{2} − 1. For further information see Abramowitz & Stegun, Chapter 22.

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

28.1 Definitions

A Chebyshev series is stored using the following structure,

     typedef struct
     {
       double * c;   /* coefficients  c[0] .. c[order] */
       int order;    /* order of expansion             */
       double a;     /* lower interval point           */
       double b;     /* upper interval point           */
       ...
     } gsl_cheb_series

The approximation is made over the range [a, b] using order+1 terms, including the coefficient c[0]. The series is computed using the following convention,

f(x) = \frac{c_{0}}{2} + ∑_{n=1} c_{n}T_{n}(x)

which is needed when accessing the coefficients directly.

28.2 Creation and Calculation of Chebyshev Series

[Function] gsl_cheb_series * gsl_cheb_alloc (const size t n)

This function allocates space for a Chebyshev series of order n and returns a pointer

     to a new gsl_cheb_series struct.

[Function] void gsl_cheb_free (gsl cheb series * cs)

This function frees a previously allocated Chebyshev series cs.

[Function] int gsl_cheb_init (gsl cheb series * cs, const gsl function * f, const

double a, const double b) This function computes the Chebyshev approximation cs for the function f over the range (a, b) to the previously specified order. The computation of the Chebyshev approximation is an O(n^{2}) process, and requires n function evaluations.

28.3 Chebyshev Series Evaluation

[Function] double gsl_cheb_eval (const gsl cheb series * cs, double x)

This function evaluates the Chebyshev series cs at a given point x.

[Function] int gsl_cheb_eval_err (const gsl cheb series * cs, const double x,

double * result, double * abserr) This function computes the Chebyshev series cs at a given point x, estimating both the series result and its absolute error abserr. The error estimate is made from the first neglected term in the series.

[Function] double gsl_cheb_eval_n (const gsl cheb series * cs, size t order,

double x) This function evaluates the Chebyshev series cs at a given point n, to (at most) the given order order.

[Function] int gsl_cheb_eval_n_err (const gsl cheb series * cs, const size t

order, const double x, double * result, double * abserr) This function evaluates a Chebyshev series cs at a given point x, estimating both the series result and its absolute error abserr, to (at most) the given order order. The error estimate is made from the first neglected term in the series.

28.4 Derivatives and Integrals

The following functions allow a Chebyshev series to be differentiated or integrated, producing a new Chebyshev series. Note that the error estimate produced by evaluating the derivative series will be underestimated due to the contribution of higher order terms being neglected.

[Function] int gsl_cheb_calc_deriv (gsl cheb series * deriv, const

gsl cheb series * cs) This function computes the derivative of the series cs, storing the derivative coefficients in the previously allocated deriv. The two series cs and deriv must have been allocated with the same order.

[Function] int gsl_cheb_calc_integ (gsl cheb series * integ, const

gsl cheb series * cs) This function computes the integral of the series cs, storing the integral coefficients in the previously allocated integ. The two series cs and integ must have been allocated with the same order. The lower limit of the integration is taken to be the left hand end of the range a.

28.5 Examples

The following example program computes Chebyshev approximations to a step function. This is an extremely difficult approximation to make, due to the discontinuity, and was chosen as an example where approximation error is visible. For smooth functions the Chebyshev approximation converges extremely rapidly and errors would not be visible.

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_chebyshev.h>
double
f (double x, void *p)

{
if (x < 0.5)
  return 0.25;
else
  return 0.75;
}
int
main (void)
{
int i, n = 10000;
gsl_cheb_series *cs = gsl_cheb_alloc (40);
gsl_function F;
F.function = f;
F.params = 0;
gsl_cheb_init (cs, &F, 0.0, 1.0);
for (i = 0; i < n; i++)
  {
    double x = i / (double)n;
    double r10 = gsl_cheb_eval_n (cs, 10, x);
    double r40 = gsl_cheb_eval (cs, x);
    printf ("%g %g %g %g\n",
            x, GSL_FN_EVAL (&F, x), r10, r40);
  }
gsl_cheb_free (cs);
return 0;
}

The output from the program gives the original function, 10-th order approximation and 40-th order approximation, all sampled at intervals of 0.001 in x.

Chapter 28: Chebyshev Approximations 307

28.6 References and Further Reading

The following paper describes the use of Chebyshev series, R. Broucke, “Ten Subroutines for the Manipulation of Chebyshev Series [C1] (Algorithm 446)”. Communications of the ACM 16(4), 254–256 (1973)

29 Series Acceleration

The functions described in this chapter accelerate the convergence of a series using the Levin u-transform. This method takes a small number of terms from the start of a series and uses a systematic approximation to compute an extrapolated value and an estimate of its error. The u-transform works for both convergent and divergent series, including asymptotic series.

These functions are declared in the header file ‘gsl_sum.h’.

29.1 Acceleration functions

The following functions compute the full Levin u-transform of a series with its error estimate. The error estimate is computed by propagating rounding errors from each term through to the final extrapolation.

These functions are intended for summing analytic series where each term is known to high accuracy, and the rounding errors are assumed to originate from finite precision. They are taken to be relative errors of order GSL_DBL_EPSILON for each term.

The calculation of the error in the extrapolated value is an O(N^{2}) process, which is expensive in time and memory. A faster but less reliable method which estimates the error from the convergence of the extrapolated value is described in the next section. For the method described here a full table of intermediate values and derivatives through to O(N) must be computed and stored, but this does give a reliable error estimate.

[Function] gsl_sum_levin_u_workspace * gsl_sum_levin_u_alloc (size t n)

This function allocates a workspace for a Levin u-transform of n terms. The size of the workspace is O(2n^{2} + 3n).

[Function] int gsl_sum_levin_u_free (gsl sum levin u workspace * w)

This function frees the memory associated with the workspace w.

[Function] int gsl_sum_levin_u_accel (const double * array, size t

array_size, gsl sum levin u workspace * w, double * sum_accel, double *

        abserr)

This function takes the terms of a series in array of size array size and computes the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in w. The extrapolated sum is stored in sum accel, with an estimate of the absolute error stored in abserr. The actual term-by-term sum is returned in w->sum_plain. The algorithm calculates the truncation error (the difference between two successive extrapolations) and round-off error (propagated from the individual terms) to choose an optimal number of terms for the extrapolation.

29.2 Acceleration functions without error estimation

The functions described in this section compute the Levin u-transform of series and attempt to estimate the error from the “truncation error” in the extrapolation, the difference between the final two approximations. Using this method avoids the need to compute an intermediate table of derivatives because the error is estimated from the behavior of the extrapolated value itself. Consequently this algorithm is an O(N) process and only requires O(N) terms of storage. If the series converges sufficiently fast then this procedure can be acceptable. It is appropriate to use this method when there is a need to compute many extrapolations of series with similar convergence properties at high-speed. For example, when numerically integrating a function defined by a parameterized series where the parameter varies only slightly. A reliable error estimate should be computed first using the full algorithm described above in order to verify the consistency of the results.

[Function] gsl_sum_levin_utrunc_workspace *
   gsl_sum_levin_utrunc_alloc (size t n)

This function allocates a workspace for a Levin u-transform of n terms, without error estimation. The size of the workspace is O(3n).

[Function] int gsl_sum_levin_utrunc_free (gsl sum levin utrunc workspace *

   w)

This function frees the memory associated with the workspace w.

[Function] int gsl_sum_levin_utrunc_accel (const double * array, size t

array_size, gsl sum levin utrunc workspace * w, double * sum_accel,

   double * abserr_trunc)

This function takes the terms of a series in array of size array size and computes the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in w. The extrapolated sum is stored in sum accel. The actual term-by-term sum is returned in w->sum_plain. The algorithm terminates when the difference between two successive extrapolations reaches a minimum or is sufficiently small. The difference between these two values is used as estimate of the error and is stored in abserr trunc. To improve the reliability of the algorithm the extrapolated values are replaced by moving averages when calculating the truncation error, smoothing out any fluctuations.

29.3 Examples

The following code calculates an estimate of ζ(2) = π^{2}/6 using the series,

ζ(2) = 1 + 1/2^{2} + 1/3^{2} + 1/4^{2} + …

After N terms the error in the sum is O(1/N), making direct summation of the series converge slowly.

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sum.h>
#define N 20
int
main (void)
{
  double t[N];
  double sum_accel, err;
  double sum = 0;

int n;
gsl_sum_levin_u_workspace * w
  = gsl_sum_levin_u_alloc (N);
const double zeta_2 = M_PI * M_PI / 6.0;
/* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */
for (n = 0; n < N; n++)
  {
    double np1 = n + 1.0;
    t[n] = 1.0 / (np1 * np1);
    sum += t[n];
  }
gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);
printf ("term-by-term sum = % .16f using %d terms\n",
        sum, N);
printf ("term-by-term sum = % .16f using %d terms\n",
        w->sum_plain, w->terms_used);
printf ("exact value      = % .16f\n", zeta_2);
printf ("accelerated sum  = % .16f using %d terms\n",
        sum_accel, w->terms_used);
printf ("estimated error  = % .16f\n", err);
printf ("actual error     = % .16f\n",
        sum_accel - zeta_2);
gsl_sum_levin_u_free (w);
return 0;
}

The output below shows that the Levin u-transform is able to obtain an estimate of the sum to 1 part in 10^{10} using the first eleven terms of the series. The error estimate returned by the function is also accurate, giving the correct number of significant digits.

$ ./a.out
term-by-term sum =  1.5961632439130233 using 20 terms
term-by-term sum =  1.5759958390005426 using 13 terms
exact value      =  1.6449340668482264
accelerated sum  =  1.6449340668166479 using 13 terms
estimated error  =  0.0000000000508580
actual error     = -0.0000000000315785

Note that a direct summation of this series would require 10^{10} terms to achieve the same precision as the accelerated sum does in 13 terms.

29.4 References and Further Reading

The algorithms used by these functions are described in the following papers, T. Fessler, W.F. Ford, D.A. Smith, hurry: An acceleration algorithm for scalar sequences and series ACM Transactions on Mathematical Software, 9(3):346–354, 1983. and Algorithm 602 9(3):355–357, 1983.

The theory of the u-transform was presented by Levin, D. Levin, Development of Non-Linear Transformations for Improving Convergence of Sequences, Intern. J. Computer Math. B3:371–388, 1973.

A review paper on the Levin Transform is available online, Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,

http://arxiv.org/abs/math/0005209.

30 Wavelet Transforms

This chapter describes functions for performing Discrete Wavelet Transforms (DWTs). The library includes wavelets for real data in both one and two dimensions. The wavelet functions are declared in the header files ‘gsl_wavelet.h’ and ‘gsl_wavelet2d.h’.

30.1 Definitions

The continuous wavelet transform and its inverse are defined by the relations,

w(s, τ) = ∫_{−∞}^{∞} f(t) ∗ ψ_{s,τ}^{∗}(t)dt

and,

f(t) = ∫_{0}^{∞} ds ∫_{−∞}^{∞} w(s, τ) ∗ ψ_{s,τ}(t)dτ

where the basis functions ψ_{s,τ} are obtained by scaling and translation from a single function, referred to as the mother wavelet.

The discrete version of the wavelet transform acts on equally-spaced samples, with fixed scaling and translation steps (s, τ). The frequency and time axes are sampled dyadically on scales of 2^{j} through a level parameter j. The resulting family of functions {ψ_{j,n}} constitutes an orthonormal basis for square-integrable signals.

The discrete wavelet transform is an O(N) algorithm, and is also referred to as the fast wavelet transform.

30.2 Initialization

The gsl_wavelet structure contains the filter coefficients defining the wavelet and any associated offset parameters.

[Function] gsl_wavelet * gsl_wavelet_alloc (const gsl wavelet type * T, size t

        k)

This function allocates and initializes a wavelet object of type T. The parameter k selects the specific member of the wavelet family. A null pointer is returned if insufficient memory is available or if a unsupported member is selected.

The following wavelet types are implemented:

[Wavelet] gsl_wavelet_daubechies
[Wavelet] gsl_wavelet_daubechies_centered

The is the Daubechies wavelet family of maximum phase with k/2 vanishing moments. The implemented wavelets are k = 4, 6, …, 20, with k even.

[Wavelet] gsl_wavelet_haar
[Wavelet] gsl_wavelet_haar_centered

This is the Haar wavelet. The only valid choice of k for the Haar wavelet is k = 2.

[Wavelet] gsl_wavelet_bspline
[Wavelet] gsl_wavelet_bspline_centered

This is the biorthogonal B-spline wavelet family of order (i, j). The implemented values of k = 100 ∗ i + j are 103, 105, 202, 204, 206, 208, 301, 303, 305 307, 309.

The centered forms of the wavelets align the coefficients of the various sub-bands on edges. Thus the resulting visualization of the coefficients of the wavelet transform in the phase plane is easier to understand.

[Function] const char * gsl_wavelet_name (const gsl wavelet * w)

This function returns a pointer to the name of the wavelet family for w.

[Function] void gsl_wavelet_free (gsl wavelet * w)

This function frees the wavelet object w.

The gsl_wavelet_workspace structure contains scratch space of the same size as the input data and is used to hold intermediate results during the transform.

[Function] gsl_wavelet_workspace * gsl_wavelet_workspace_alloc (size t
   n)

This function allocates a workspace for the discrete wavelet transform. To perform a one-dimensional transform on n elements, a workspace of size n must be provided. For two-dimensional transforms of n-by-n matrices it is sufficient to allocate a workspace of size n, since the transform operates on individual rows and columns.

[Function] void gsl_wavelet_workspace_free (gsl wavelet workspace * work)

This function frees the allocated workspace work.

30.3 Transform Functions

This sections describes the actual functions performing the discrete wavelet transform. Note that the transforms use periodic boundary conditions. If the signal is not periodic in the sample length then spurious coefficients will appear at the beginning and end of each level of the transform.

30.3.1 Wavelet transforms in one dimension

[Function] int gsl_wavelet_transform (const gsl wavelet * w, double * data,

size t stride, size t n, gsl wavelet direction dir, gsl wavelet workspace *

   work)

[Function] int gsl_wavelet_transform_forward (const gsl wavelet * w, double *

data, size t stride, size t n, gsl wavelet workspace * work)

[Function] int gsl_wavelet_transform_inverse (const gsl wavelet * w, double *

data, size t stride, size t n, gsl wavelet workspace * work) These functions compute in-place forward and inverse discrete wavelet transforms of length n with stride stride on the array data. The length of the transform n is restricted to powers of two. For the transform version of the function the argument dir can be either forward (+1) or backward (−1). A workspace work of length n must be provided. For the forward transform, the elements of the original array are replaced by the discrete wavelet transform f_{i} → w_{j,k} in a packed triangular storage layout, where j is the index of the level j = 0 … J − 1 and k is the index of the coefficient within each level, k = 0 … 2^{j} − 1. The total number of levels is J = log_{2}(n). The output data has the following form,

(s_{−1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, ⋯, d_{j,k}, ⋯, d_{J−1,2^{J−1}−1})

where the first element is the smoothing coefficient s_{−1,0}, followed by the detail coefficients d_{j,k} for each level j. The backward transform inverts these coefficients to obtain the original data. These functions return a status of GSL_SUCCESS upon successful completion. GSL_ EINVAL is returned if n is not an integer power of 2 or if insufficient workspace is provided.

30.3.2 Wavelet transforms in two dimension

The library provides functions to perform two-dimensional discrete wavelet transforms on square matrices. The matrix dimensions must be an integer power of two. There are two possible orderings of the rows and columns in the two-dimensional wavelet transform, referred to as the “standard” and “non-standard” forms.

The “standard” transform performs a complete discrete wavelet transform on the rows of the matrix, followed by a separate complete discrete wavelet transform on the columns of the resulting row-transformed matrix. This procedure uses the same ordering as a twodimensional fourier transform.

The “non-standard” transform is performed in interleaved passes on the rows and columns of the matrix for each level of the transform. The first level of the transform is applied to the matrix rows, and then to the matrix columns. This procedure is then repeated across the rows and columns of the data for the subsequent levels of the transform, until the full discrete wavelet transform is complete. The non-standard form of the discrete wavelet transform is typically used in image analysis.

The functions described in this section are declared in the header file ‘gsl_wavelet2d.h’.

[Function] int gsl_wavelet2d_transform (const gsl wavelet * w, double * data,

size t tda, size t size1, size t size2, gsl wavelet direction dir,

gsl wavelet workspace * work)

[Function] int gsl_wavelet2d_transform_forward (const gsl wavelet * w,

double * data, size t tda, size t size1, size t size2, gsl wavelet workspace *

   work)
[Function] int gsl_wavelet2d_transform_inverse (const gsl wavelet * w,

double * data, size t tda, size t size1, size t size2, gsl wavelet workspace *

   work)

These functions compute two-dimensional in-place forward and inverse discrete wavelet transforms in standard and non-standard forms on the array data stored in row-major form with dimensions size1 and size2 and physical row length tda. The dimensions must be equal (square matrix) and are restricted to powers of two. For the transform version of the function the argument dir can be either forward (+1) or backward (−1). A workspace work of the appropriate size must be provided. On exit, the appropriate elements of the array data are replaced by their two-dimensional wavelet transform. The functions return a status of GSL_SUCCESS upon successful completion. GSL_ EINVAL is returned if size1 and size2 are not equal and integer powers of 2, or if insufficient workspace is provided.

[Function] int gsl_wavelet2d_transform_matrix (const gsl wavelet * w,

gsl matrix * m, gsl wavelet direction dir, gsl wavelet workspace * work)

[Function] int gsl_wavelet2d_transform_matrix_forward (const gsl wavelet *

w, gsl matrix * m, gsl wavelet workspace * work)

[Function] int gsl_wavelet2d_transform_matrix_inverse (const gsl wavelet *

w, gsl matrix * m, gsl wavelet workspace * work) These functions compute the two-dimensional in-place wavelet transform on a matrix a.

[Function] int gsl_wavelet2d_nstransform (const gsl wavelet * w, double *

data, size t tda, size t size1, size t size2, gsl wavelet direction dir, gsl wavelet workspace * work)

[Function] int gsl_wavelet2d_nstransform_forward (const gsl wavelet * w,

double * data, size t tda, size t size1, size t size2, gsl wavelet workspace *

   work)
[Function] int gsl_wavelet2d_nstransform_inverse (const gsl wavelet * w,

double * data, size t tda, size t size1, size t size2, gsl wavelet workspace *

   work)

These functions compute the two-dimensional wavelet transform in non-standard form.

[Function] int gsl_wavelet2d_nstransform_matrix (const gsl wavelet * w,

gsl matrix * m, gsl wavelet direction dir, gsl wavelet workspace * work)

[Function] int gsl_wavelet2d_nstransform_matrix_forward (const

gsl wavelet * w, gsl matrix * m, gsl wavelet workspace * work)

[Function] int gsl_wavelet2d_nstransform_matrix_inverse (const

gsl wavelet * w, gsl matrix * m, gsl wavelet workspace * work) These functions compute the non-standard form of the two-dimensional in-place wavelet transform on a matrix a.

30.4 Examples

The following program demonstrates the use of the one-dimensional wavelet transform functions. It computes an approximation to an input signal (of length 256) using the 20 largest components of the wavelet transform, while setting the others to zero.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_wavelet.h>
int
main (int argc, char **argv)
{
  int i, n = 256, nc = 20;
  double *data = malloc (n * sizeof (double));
  double *abscoeff = malloc (n * sizeof (double));
  size_t *p = malloc (n * sizeof (size_t));
  gsl_wavelet *w;
  gsl_wavelet_workspace *work;

w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
work = gsl_wavelet_workspace_alloc (n);
FILE *f = fopen (argv[1], "r");
for (i = 0; i < n; i++)
  {
    fscanf (f, "%lg", &data[i]);
  }
fclose (f);
gsl_wavelet_transform_forward (w, data, 1, n, work);
for (i = 0; i < n; i++)
  {
    abscoeff[i] = fabs (data[i]);
  }
gsl_sort_index (p, abscoeff, 1, n);
for (i = 0; (i + nc) < n; i++)
  data[p[i]] = 0;
gsl_wavelet_transform_inverse (w, data, 1, n, work);
for (i = 0; i < n; i++)
  {
    printf ("%g\n", data[i]);
  }
}

The output can be used with the gnu plotutils graph program,

$ ./a.out ecg.dat > dwt.dat
$ graph -T ps -x 0 256 32 -h 0.3 -a dwt.dat > dwt.ps

The graphs below show an original and compressed version of a sample ECG recording from the MIT-BIH Arrhythmia Database, part of the PhysioNet archive of public-domain of medical datasets.

Original (upper) and wavelet-compressed (lower) ECG signals, using the 20 largest components of the Daubechies(4) discrete wavelet transform.

30.5 References and Further Reading

The mathematical background to wavelet transforms is covered in the original lectures by Daubechies,

Ingrid Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics (1992), SIAM, ISBN 0898712742.

An easy to read introduction to the subject with an emphasis on the application of the wavelet transform in various branches of science is,

Paul S. Addison. The Illustrated Wavelet Transform Handbook. Institute of Physics Publishing (2002), ISBN 0750306920.

For extensive coverage of signal analysis by wavelets, wavelet packets and local cosine bases see,

S. G. Mallat. A wavelet tour of signal processing (Second edition). Academic Press (1999), ISBN 012466606X.

The concept of multiresolution analysis underlying the wavelet transform is described in, S. G. Mallat. Multiresolution Approximations and Wavelet Orthonormal Bases of L^{2}(R). Transactions of the American Mathematical Society, 315(1), 1989, 69–87. S. G. Mallat. A Theory for Multiresolution Signal Decomposition—The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 1989, 674–693.

The coefficients for the individual wavelet families implemented by the library can be found in the following papers,

I. Daubechies. Orthonormal Bases of Compactly Supported Wavelets. Communications on Pure and Applied Mathematics, 41 (1988) 909–996.

A. Cohen, I. Daubechies, and J.-C. Feauveau. Biorthogonal Bases of Compactly Supported Wavelets. Communications on Pure and Applied Mathematics, 45 (1992) 485– 560.

The PhysioNet archive of physiological datasets can be found online at http://www.physionet.org/ and is described in the following paper,

Goldberger et al. PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation 101(23):e215-e220

31 Discrete Hankel Transforms

This chapter describes functions for performing Discrete Hankel Transforms (DHTs). The functions are declared in the header file ‘gsl_dht.h’.

31.1 Definitions

The discrete Hankel transform acts on a vector of sampled data, where the samples are assumed to have been taken at points related to the zeroes of a Bessel function of fixed order; compare this to the case of the discrete Fourier transform, where samples are taken at points related to the zeroes of the sine or cosine function.

Specifically, let f(t) be a function on the unit interval. Then the finite ν-Hankel transform of f(t) is defined to be the set of numbers g_{m} given by,

g_{m} = ∫_{0}^{1} tdt J_{ν}(j_{ν,m}t)f(t),

so that,

f(t) = ∑_{m=1}^{∞} \frac{2J_{ν}(j_{ν,m}x)}{J_{ν+1}(j_{ν,m})^{2}} g_{m}.

Suppose that f is band-limited in the sense that g_{m} = 0 for m > M. Then we have the following fundamental sampling theorem.

g_{m} = \frac{2}{j_{ν,M}^{2}} ∑_{k=1}^{M−1} f (\frac{j_{ν,k}}{j_{ν,M}} ) \frac{J_{ν}(j_{ν,m}j_{ν,k}/j_{ν,M})}{J_{ν+1}(j_{ν,k})^{2}}.

It is this discrete expression which defines the discrete Hankel transform. The kernel in the summation above defines the matrix of the ν-Hankel transform of size M − 1. The coefficients of this matrix, being dependent on ν and M, must be precomputed and stored; the gsl_dht object encapsulates this data. The allocation function gsl_dht_alloc returns a gsl_dht object which must be properly initialized with gsl_dht_init before it can be used to perform transforms on data sample vectors, for fixed ν and M, using the gsl_ dht_apply function. The implementation allows a scaling of the fundamental interval, for convenience, so that one can assume the function is defined on the interval [0, X], rather than the unit interval.

Notice that by assumption f(t) vanishes at the endpoints of the interval, consistent with the inversion formula and the sampling formula given above. Therefore, this transform corresponds to an orthogonal expansion in eigenfunctions of the Dirichlet problem for the Bessel differential equation.

31.2 Functions

[Function] gsl_dht * gsl_dht_alloc (size t size)

This function allocates a Discrete Hankel transform object of size size.

[Function] int gsl_dht_init (gsl dht * t, double nu, double xmax)

This function initializes the transform t for the given values of nu and x.

[Function] gsl_dht * gsl_dht_new (size t size, double nu, double xmax)

This function allocates a Discrete Hankel transform object of size size and initializes it for the given values of nu and x.

[Function] void gsl_dht_free (gsl dht * t)

This function frees the transform t.

[Function] int gsl_dht_apply (const gsl dht * t, double * f_in, double * f_out)

This function applies the transform t to the array f in whose size is equal to the size of the transform. The result is stored in the array f out which must be of the same length.

[Function] double gsl_dht_x_sample (const gsl dht * t, int n)

This function returns the value of the n-th sample point in the unit interval, (j_{ν,n+1}/j_{ν,M})X. These are the points where the function f(t) is assumed to be sampled.

[Function] double gsl_dht_k_sample (const gsl dht * t, int n)

This function returns the value of the n-th sample point in “k-space”, j_{ν,n+1}/X.

31.3 References and Further Reading

The algorithms used by these functions are described in the following papers, H. Fisk Johnson, Comp. Phys. Comm. 43, 181 (1987). D. Lemoine, J. Chem. Phys. 101, 3936 (1994).

32 One dimensional Root-Finding

This chapter describes routines for finding roots of arbitrary one-dimensional functions. The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs.

The header file ‘gsl_roots.h’ contains prototypes for the root finding functions and related declarations.

32.1 Overview

One-dimensional root finding algorithms can be divided into two classes, root bracketing and root polishing. Algorithms which proceed by bracketing a root are guaranteed to converge. Bracketing algorithms begin with a bounded region known to contain a root. The size of this bounded region is reduced, iteratively, until it encloses the root to a desired tolerance. This provides a rigorous error estimate for the location of the root.

The technique of root polishing attempts to improve an initial guess to the root. These algorithms converge only if started “close enough” to a root, and sacrifice a rigorous error bound for speed. By approximating the behavior of a function in the vicinity of a root they attempt to find a higher order improvement of an initial guess. When the behavior of the function is compatible with the algorithm and a good initial guess is available a polishing algorithm can provide rapid convergence.

In GSL both types of algorithm are available in similar frameworks. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

• initialize solver state, s, for algorithm T

• update s using the iteration T

• test s for convergence, and repeat iteration if necessary

The state for bracketing solvers is held in a gsl_root_fsolver struct. The updating procedure uses only function evaluations (not derivatives). The state for root polishing solvers is held in a gsl_root_fdfsolver struct. The updates require both the function and its derivative (hence the name fdf) to be supplied by the user.

32.2 Caveats

Note that root finding functions can only search for one root at a time. When there are several roots in the search area, the first root to be found will be returned; however it is difficult to predict which of the roots this will be. In most cases, no error will be reported if you try to find a root in an area where there is more than one.

Care must be taken when a function may have a multiple root (such as f(x) = (x−x_{0})^{2} or f(x) = (x − x_{0})^{3}). It is not possible to use root-bracketing algorithms on even-multiplicity roots. For these algorithms the initial interval must contain a zero-crossing, where the function is negative at one end of the interval and positive at the other end. Roots with even-multiplicity do not cross zero, but only touch it instantaneously. Algorithms based on root bracketing will still work for odd-multiplicity roots (e.g. cubic, quintic, …). Root polishing algorithms generally work with higher multiplicity roots, but at a reduced rate of convergence. In these cases the Steffenson algorithm can be used to accelerate the convergence of multiple roots.

While it is not absolutely required that f have a root within the search region, numerical root finding functions should not be used haphazardly to check for the existence of roots. There are better ways to do this. Because it is easy to create situations where numerical root finders can fail, it is a bad idea to throw a root finder at a function you do not know much about. In general it is best to examine the function visually by plotting before searching for a root.

32.3 Initializing the Solver

[Function] gsl_root_fsolver * gsl_root_fsolver_alloc (const

gsl root fsolver type * T) This function returns a pointer to a newly allocated instance of a solver of type T. For example, the following code creates an instance of a bisection solver,

          const gsl_root_fsolver_type * T
            = gsl_root_fsolver_bisection;
          gsl_root_fsolver * s
            = gsl_root_fsolver_alloc (T);

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] gsl_root_fdfsolver * gsl_root_fdfsolver_alloc (const

gsl root fdfsolver type * T) This function returns a pointer to a newly allocated instance of a derivative-based solver of type T. For example, the following code creates an instance of a Newton-Raphson solver,

          const gsl_root_fdfsolver_type * T
            = gsl_root_fdfsolver_newton;
          gsl_root_fdfsolver * s
            = gsl_root_fdfsolver_alloc (T);

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] int gsl_root_fsolver_set (gsl root fsolver * s, gsl function * f,

        double x_lower, double x_upper)

This function initializes, or reinitializes, an existing solver s to use the function f and the initial search interval [x lower, x upper].

[Function] int gsl_root_fdfsolver_set (gsl root fdfsolver * s, gsl function fdf *

fdf, double root) This function initializes, or reinitializes, an existing solver s to use the function and derivative fdf and the initial guess root.

[Function] void gsl_root_fsolver_free (gsl root fsolver * s)

[Function] void gsl_root_fdfsolver_free (gsl root fdfsolver * s)

These functions free all the memory associated with the solver s.

[Function] const char * gsl_root_fsolver_name (const gsl root fsolver * s)
[Function] const char * gsl_root_fdfsolver_name (const gsl root fdfsolver *
   s)

These functions return a pointer to the name of the solver. For example,

     printf ("s is a '%s' solver\n",
             gsl_root_fsolver_name (s));

would print something like s is a 'bisection' solver.

32.4 Providing the function to solve

You must provide a continuous function of one variable for the root finders to operate on, and, sometimes, its first derivative. In order to allow for general parameters the functions are defined by the following data types:

[Data Type] gsl_function

This data type defines a general function with parameters.

double (* function) (double x, void * params)

this function should return the value f(x, params) for argument x and parameters params

void * params

a pointer to the parameters of the function

Here is an example for the general quadratic function,

f(x) = ax^{2} + bx + c

with a = 3, b = 2, c = 1. The following code defines a gsl_function F which you could pass to a root finder:

struct my_f_params { double a; double b; double c; };
double
my_f (double x, void * p) {
   struct my_f_params * params
     = (struct my_f_params *)p;
   double a = (params->a);
   double b = (params->b);
   double c = (params->c);
   return  (a * x + b) * x + c;
}
gsl_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };

F.function = &my_f;
F.params = &params;

The function f(x) can be evaluated using the following macro,

#define GSL_FN_EVAL(F,x)
    (*((F)->function))(x,(F)->params)
[Data Type] gsl_function_fdf

This data type defines a general function with parameters and its first derivative.

double (* f) (double x, void * params)

this function should return the value of f(x, params) for argument x and parameters params

double (* df) (double x, void * params)

this function should return the value of the derivative of f with respect to x, f′(x, params), for argument x and parameters params

void (* fdf) (double x, void * params, double * f, double * df)

this function should set the values of the function f to f(x, params) and its derivative df to f′(x, params) for argument x and parameters params. This function provides an optimization of the separate functions for f(x) and f′(x)—it is always faster to compute the function and its derivative at the same time.

void * params

a pointer to the parameters of the function

Here is an example where f(x) = exp(2x):

double
my_f (double x, void * params)
{
   return exp (2 * x);
}
double
my_df (double x, void * params)
{
   return 2 * exp (2 * x);
}
void
my_fdf (double x, void * params,
        double * f, double * df)
{
   double t = exp (2 * x);
   *f = t;
   *df = 2 * t;   /* uses existing value */
}

     gsl_function_fdf FDF;
     FDF.f = &my_f;
     FDF.df = &my_df;
     FDF.fdf = &my_fdf;
     FDF.params = 0;

The function f(x) can be evaluated using the following macro,

     #define GSL_FN_FDF_EVAL_F(FDF,x)
          (*((FDF)->f))(x,(FDF)->params)

The derivative f′(x) can be evaluated using the following macro,

     #define GSL_FN_FDF_EVAL_DF(FDF,x)
          (*((FDF)->df))(x,(FDF)->params)

and both the function y = f(x) and its derivative dy = f′(x) can be evaluated at the same time using the following macro,

     #define GSL_FN_FDF_EVAL_F_DF(FDF,x,y,dy)
          (*((FDF)->fdf))(x,(FDF)->params,(y),(dy))

The macro stores f(x) in its y argument and f′(x) in its dy argument—both of these should be pointers to double.

32.5 Search Bounds and Guesses

You provide either search bounds or an initial guess; this section explains how search bounds and guesses work and how function arguments control them.

A guess is simply an x value which is iterated until it is within the desired precision of a root. It takes the form of a double.

Search bounds are the endpoints of a interval which is iterated until the length of the interval is smaller than the requested precision. The interval is defined by two values, the lower limit and the upper limit. Whether the endpoints are intended to be included in the interval or not depends on the context in which the interval is used.

32.6 Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code.

[Function] int gsl_root_fsolver_iterate (gsl root fsolver * s)
[Function] int gsl_root_fdfsolver_iterate (gsl root fdfsolver * s)

These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned,

     GSL_EBADFUNC

the iteration encountered a singular point where the function or its derivative evaluated to Inf or NaN.

GSL_EZERODIV

the derivative of the function vanished at the iteration point, preventing the algorithm from continuing without a division by zero.

The solver maintains a current best estimate of the root at all times. The bracketing solvers also keep track of the current best interval bounding the root. This information can be accessed with the following auxiliary functions,

[Function] double gsl_root_fsolver_root (const gsl root fsolver * s)

[Function] double gsl_root_fdfsolver_root (const gsl root fdfsolver * s)

These functions return the current estimate of the root for the solver s.

[Function] double gsl_root_fsolver_x_lower (const gsl root fsolver * s)

[Function] double gsl_root_fsolver_x_upper (const gsl root fsolver * s)

These functions return the current bracketing interval for the solver s.

32.7 Search Stopping Parameters

A root finding procedure should stop when one of the following conditions is true: • A root has been found to within the user-specified precision.

• A user-specified maximum number of iterations has been reached.

• An error has occurred.

The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result in several standard ways.

[Function] int gsl_root_test_interval (double x_lower, double x_upper,

double epsabs, double epsrel) This function tests for the convergence of the interval [x lower, x upper] with absolute error epsabs and relative error epsrel. The test returns GSL_SUCCESS if the following condition is achieved,

|a − b| < epsabs + epsrel min(|a|, |b|)

when the interval x = [a, b] does not include the origin. If the interval includes the origin then min(|a|, |b|) is replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error is accurately estimated for roots close to the origin. This condition on the interval also implies that any estimate of the root r in the interval satisfies the same condition with respect to the true root r^{∗},

|r − r^{∗}| < epsabs + epsrel r^{∗}

assuming that the true root r^{∗} is contained within the interval.

[Function] int gsl_root_test_delta (double x1, double x0, double epsabs,

double epsrel) This function tests for the convergence of the sequence …, x0, x1 with absolute error epsabs and relative error epsrel. The test returns GSL_SUCCESS if the following condition is achieved,

|x_{1} − x_{0}| < epsabs + epsrel |x_{1}|

and returns GSL_CONTINUE otherwise.

[Function] int gsl_root_test_residual (double f, double epsabs)

This function tests the residual value f against the absolute error bound epsabs. The test returns GSL_SUCCESS if the following condition is achieved,

|f| < epsabs

and returns GSL_CONTINUE otherwise. This criterion is suitable for situations where the precise location of the root, x, is unimportant provided a value can be found where the residual, |f(x)|, is small enough.

32.8 Root Bracketing Algorithms

The root bracketing algorithms described in this section require an initial interval which is guaranteed to contain a root—if a and b are the endpoints of the interval then f(a) must differ in sign from f(b). This ensures that the function crosses zero at least once in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.

Note that a bracketing algorithm cannot find roots of even degree, since these do not cross the x-axis.

[Solver] gsl_root_fsolver_bisection

The bisection algorithm is the simplest method of bracketing the roots of a function. It is the slowest algorithm provided by the library, with linear convergence. On each iteration, the interval is bisected and the value of the function at the midpoint is calculated. The sign of this value is used to determine which half of the interval does not contain a root. That half is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small. At any time the current estimate of the root is taken as the midpoint of the interval.

[Solver] gsl_root_fsolver_falsepos

The false position algorithm is a method of finding roots based on linear interpolation. Its convergence is linear, but it is usually faster than bisection. On each iteration a line is drawn between the endpoints (a, f(a)) and (b, f(b)) and the point where this line crosses the x-axis taken as a “midpoint”. The value of the function at this point is calculated and its sign is used to determine which side of the interval does not contain a root. That side is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small. The best estimate of the root is taken from the linear interpolation of the interval on the current iteration.

[Solver] gsl_root_fsolver_brent

The Brent-Dekker method (referred to here as Brent’s method) combines an interpolation strategy with the bisection algorithm. This produces a fast algorithm which is still robust.

On each iteration Brent’s method approximates the function using an interpolating curve. On the first iteration this is a linear interpolation of the two endpoints. For subsequent iterations the algorithm uses an inverse quadratic fit to the last three points, for higher accuracy. The intercept of the interpolating curve with the x-axis is taken as a guess for the root. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary bisection step. The best estimate of the root is taken from the most recent interpolation or bisection.

32.9 Root Finding Algorithms using Derivatives

The root polishing algorithms described in this section require an initial guess for the location of the root. There is no absolute guarantee of convergence—the function must be suitable for this technique and the initial guess must be sufficiently close to the root for it to work. When these conditions are satisfied then convergence is quadratic.

These algorithms make use of both the function and its derivative.

[Derivative Solver] gsl_root_fdfsolver_newton

Newton’s Method is the standard root-polishing algorithm. The algorithm begins with an initial guess for the location of the root. On each iteration, a line tangent to the function f is drawn at that position. The point where this line crosses the x-axis becomes the new guess. The iteration is defined by the following sequence,

x_{i+1} = x_{i} − \frac{f(x_{i})}{f′(x_{i})}

Newton’s method converges quadratically for single roots, and linearly for multiple roots.

[Derivative Solver] gsl_root_fdfsolver_secant

The secant method is a simplified version of Newton’s method which does not require the computation of the derivative on every step. On its first iteration the algorithm begins with Newton’s method, using the derivative to compute a first step,

x_{1} = x_{0} − \frac{f(x_{0})}{f′(x_{0})}

Subsequent iterations avoid the evaluation of the derivative by replacing it with a numerical estimate, the slope of the line through the previous two points,

x_{i+1} = x_{i} − \frac{f(x_{i})}{f′_{est}} where f′_{est} = \frac{f(x_{i}) − f(x_{i−1})}{x_{i} − x_{i−1}}

When the derivative does not change significantly in the vicinity of the root the secant method gives a useful saving. Asymptotically the secant method is faster than Newton’s method whenever the cost of evaluating the derivative is more than 0.44 times the cost of evaluating the function itself. As with all methods of computing a numerical derivative the estimate can suffer from cancellation errors if the separation of the points becomes too small.

On single roots, the method has a convergence of order (1 + \sqrt{5})/2 (approximately 1.62). It converges linearly for multiple roots.

[Derivative Solver] gsl_root_fdfsolver_steffenson

The Steffenson Method provides the fastest convergence of all the routines. It combines the basic Newton algorithm with an Aitken “delta-squared” acceleration. If the Newton iterates are x_{i} then the acceleration procedure generates a new sequence R_{i},

R_{i} = x_{i} − \frac{(x_{i+1} − x_{i})^{2}}{(x_{i+2} − 2x_{i+1} + x_{i})}

which converges faster than the original sequence under reasonable conditions. The new sequence requires three terms before it can produce its first value so the method returns accelerated values on the second and subsequent iterations. On the first iteration it returns the ordinary Newton estimate. The Newton iterate is also returned if the denominator of the acceleration term ever becomes zero. As with all acceleration procedures this method can become unstable if the function is not well-behaved.

32.10 Examples

For any root finding algorithm we need to prepare the function to be solved. For this example we will use the general quadratic equation described earlier. We first need a header file (‘demo_fn.h’) to define the function parameters,

struct quadratic_params
  {
    double a, b, c;
  };
double quadratic (double x, void *params);
double quadratic_deriv (double x, void *params);
void quadratic_fdf (double x, void *params,
                    double *y, double *dy);

We place the function definitions in a separate file (‘demo_fn.c’),

double
quadratic (double x, void *params)
{
  struct quadratic_params *p
    = (struct quadratic_params *) params;
  double a = p->a;
  double b = p->b;
  double c = p->c;
  return (a * x + b) * x + c;
}
double

quadratic_deriv (double x, void *params)
{
  struct quadratic_params *p
    = (struct quadratic_params *) params;
  double a = p->a;
  double b = p->b;
  double c = p->c;
  return 2.0 * a * x + b;
}
void
quadratic_fdf (double x, void *params,
               double *y, double *dy)
{
  struct quadratic_params *p
    = (struct quadratic_params *) params;
  double a = p->a;
  double b = p->b;
  double c = p->c;
  *y = (a * x + b) * x + c;
  *dy = 2.0 * a * x + b;
}

The first program uses the function solver gsl_root_fsolver_brent for Brent’s method and the general quadratic defined above to solve the following equation,

x^{2} − 5 = 0

with solution x = \sqrt{5} = 2.236068...

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>
#include "demo_fn.h"
#include "demo_fn.c"
int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_root_fsolver_type *T;
  gsl_root_fsolver *s;

double r = 0, r_expected = sqrt (5.0);
double x_lo = 0.0, x_hi = 5.0;
gsl_function F;
struct quadratic_params params = {1.0, 0.0, -5.0};
F.function = &quadratic;
F.params = &params;
T = gsl_root_fsolver_brent;
s = gsl_root_fsolver_alloc (T);
gsl_root_fsolver_set (s, &F, x_lo, x_hi);
printf ("using %s method\n",
        gsl_root_fsolver_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n",
        "iter", "lower", "upper", "root",
        "err", "err(est)");
do
  {
    iter++;
    status = gsl_root_fsolver_iterate (s);
    r = gsl_root_fsolver_root (s);
    x_lo = gsl_root_fsolver_x_lower (s);
    x_hi = gsl_root_fsolver_x_upper (s);
    status = gsl_root_test_interval (x_lo, x_hi,
                                     0, 0.001);
    if (status == GSL_SUCCESS)
      printf ("Converged:\n");
    printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
            iter, x_lo, x_hi,
            r, r - r_expected,
            x_hi - x_lo);
  }
while (status == GSL_CONTINUE && iter < max_iter);
return status;
}

Here are the results of the iterations,

$ ./a.out
using brent method
iter [    lower,     upper]      root        err  err(est)
  1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
  2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
  3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
  4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000

  5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
Converged:
  6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666

If the program is modified to use the bisection solver instead of Brent’s method, by changing

gsl_root_fsolver_brent to gsl_root_fsolver_bisection the slower convergence of the

Bisection method can be observed,

$ ./a.out
using bisection method
iter [    lower,     upper]      root        err  err(est)
  1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
  2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
  3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
  4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
  5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
  6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
  7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
  8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
  9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
 10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
 11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
Converged:
 12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207

The next program solves the same function using a derivative solver instead.

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>
#include "demo_fn.h"
#include "demo_fn.c"
int
main (void)
{
int status;
int iter = 0, max_iter = 100;
const gsl_root_fdfsolver_type *T;
gsl_root_fdfsolver *s;
double x0, x = 5.0, r_expected = sqrt (5.0);
gsl_function_fdf FDF;
struct quadratic_params params = {1.0, 0.0, -5.0};
FDF.f = &quadratic;
FDF.df = &quadratic_deriv;
FDF.fdf = &quadratic_fdf;
FDF.params = &params;
T = gsl_root_fdfsolver_newton;
s = gsl_root_fdfsolver_alloc (T);
gsl_root_fdfsolver_set (s, &FDF, x);

       printf ("using %s method\n",
               gsl_root_fdfsolver_name (s));
       printf ("%-5s %10s %10s %10s\n",
               "iter", "root", "err", "err(est)");
       do
         {
           iter++;
           status = gsl_root_fdfsolver_iterate (s);
           x0 = x;
           x = gsl_root_fdfsolver_root (s);
           status = gsl_root_test_delta (x, x0, 0, 1e-3);
           if (status == GSL_SUCCESS)
             printf ("Converged:\n");
           printf ("%5d %10.7f %+10.7f %10.7f\n",
                   iter, x, x - r_expected, x - x0);
         }
       while (status == GSL_CONTINUE && iter < max_iter);
       return status;
     }

Here are the results for Newton’s method,

     $ ./a.out
     using newton method
     iter        root        err   err(est)
         1  3.0000000 +0.7639320 -2.0000000
         2  2.3333333 +0.0972654 -0.6666667
         3  2.2380952 +0.0020273 -0.0952381
     Converged:
         4  2.2360689 +0.0000009 -0.0020263

Note that the error can be estimated more accurately by taking the difference between the current iterate and next iterate rather than the previous iterate. The other derivative solvers

can be investigated by changing gsl_root_fdfsolver_newton to gsl_root_fdfsolver_
secant or gsl_root_fdfsolver_steffenson.

32.11 References and Further Reading

For information on the Brent-Dekker algorithm see the following two papers, R. P. Brent, “An algorithm with guaranteed convergence for finding a zero of a function”, Computer Journal, 14 (1971) 422–425 J. C. P. Bus and T. J. Dekker, “Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function”, ACM Transactions of Mathematical Software, Vol. 1 No. 4 (1975) 330–345

33 One dimensional Minimization

This chapter describes routines for finding minima of arbitrary one-dimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs.

The header file ‘gsl_min.h’ contains prototypes for the minimization functions and related declarations. To use the minimization algorithms to find the maximum of a function simply invert its sign.

33.1 Overview

The minimization algorithms begin with a bounded region known to contain a minimum. The region is described by a lower bound a and an upper bound b, with an estimate of the location of the minimum x.

The value of the function at x must be less than the value of the function at the ends of the interval,

f(a) > f(x) < f(b)

This condition guarantees that a minimum is contained somewhere within the interval. On each iteration a new point x′ is selected using one of the available algorithms. If the new point is a better estimate of the minimum, i.e. where f(x′) < f(x), then the current estimate of the minimum x is updated. The new point also allows the size of the bounded interval to be reduced, by choosing the most compact set of points which satisfies the constraint f(a) > f(x) < f(b). The interval is reduced until it encloses the true minimum to a desired tolerance. This provides a best estimate of the location of the minimum and a rigorous error estimate.

Several bracketing algorithms are available within a single framework. The user provides a high-level driver for the algorithm, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

• initialize minimizer state, s, for algorithm T • update s using the iteration T

• test s for convergence, and repeat iteration if necessary

The state for the minimizers is held in a gsl_min_fminimizer struct. The updating procedure uses only function evaluations (not derivatives).

33.2 Caveats

Note that minimization functions can only search for one minimum at a time. When there are several minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a minimum in an area where there is more than one.

With all minimization algorithms it can be difficult to determine the location of the minimum to full numerical precision. The behavior of the function in the region of the minimum x^{∗} can be approximated by a Taylor expansion,

y = f(x^{∗}) + \frac{1}{2}f′′(x^{∗})(x − x^{∗})^{2}

and the second term of this expansion can be lost when added to the first term at finite precision. This magnifies the error in locating x^{∗}, making it proportional to \sqrt{ϵ} (where ϵ is the relative accuracy of the floating point numbers). For functions with higher order minima, such as x^{4}, the magnification of the error is correspondingly worse. The best that can be achieved is to converge to the limit of numerical accuracy in the function values, rather than the location of the minimum itself.

33.3 Initializing the Minimizer

[Function] gsl_min_fminimizer * gsl_min_fminimizer_alloc (const

gsl min fminimizer type * T) This function returns a pointer to a newly allocated instance of a minimizer of type T. For example, the following code creates an instance of a golden section minimizer,

          const gsl_min_fminimizer_type * T
            = gsl_min_fminimizer_goldensection;
          gsl_min_fminimizer * s
            = gsl_min_fminimizer_alloc (T);

If there is insufficient memory to create the minimizer then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] int gsl_min_fminimizer_set (gsl min fminimizer * s, gsl function *

        f, double x_minimum, double x_lower, double x_upper)

This function sets, or resets, an existing minimizer s to use the function f and the initial search interval [x lower, x upper], with a guess for the location of the minimum x minimum.

If the interval given does not contain a minimum, then the function returns an error

     code of GSL_EINVAL.

[Function] int gsl_min_fminimizer_set_with_values (gsl min fminimizer * s,

gsl function * f, double x_minimum, double f_minimum, double x_lower,

   double f_lower, double x_upper, double f_upper)

This function is equivalent to gsl_min_fminimizer_set but uses the values f minimum, f lower and f upper instead of computing f(x_minimum), f(x_lower)

and f(x_upper).

[Function] void gsl_min_fminimizer_free (gsl min fminimizer * s)

This function frees all the memory associated with the minimizer s.

[Function] const char * gsl_min_fminimizer_name (const gsl min fminimizer *
   s)

This function returns a pointer to the name of the minimizer. For example,

     printf ("s is a '%s' minimizer\n",
             gsl_min_fminimizer_name (s));

would print something like s is a 'brent' minimizer.

33.4 Providing the function to minimize

You must provide a continuous function of one variable for the minimizers to operate on. In order to allow for general parameters the functions are defined by a gsl_function data type (see Section 32.4 [Providing the function to solve], page 323).

33.5 Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any minimizer of the corresponding type. The same functions work for all minimizers so that different methods can be substituted at runtime without modifications to the code.

[Function] int gsl_min_fminimizer_iterate (gsl min fminimizer * s)

This function performs a single iteration of the minimizer s. If the iteration encounters an unexpected problem then an error code will be returned,

GSL_EBADFUNC

the iteration encountered a singular point where the function evaluated

          to Inf or NaN.
GSL_FAILURE

the algorithm could not improve the current best approximation or bounding interval.

The minimizer maintains a current best estimate of the position of the minimum at all times, and the current interval bounding the minimum. This information can be accessed with the following auxiliary functions,

[Function] double gsl_min_fminimizer_x_minimum (const gsl min fminimizer *
   s)

This function returns the current estimate of the position of the minimum for the minimizer s.

[Function] double gsl_min_fminimizer_x_upper (const gsl min fminimizer * s)

[Function] double gsl_min_fminimizer_x_lower (const gsl min fminimizer * s)

These functions return the current upper and lower bound of the interval for the minimizer s.

[Function] double gsl_min_fminimizer_f_minimum (const gsl min fminimizer *
   s)

[Function] double gsl_min_fminimizer_f_upper (const gsl min fminimizer * s)

[Function] double gsl_min_fminimizer_f_lower (const gsl min fminimizer * s)

These functions return the value of the function at the current estimate of the minimum and at the upper and lower bounds of the interval for the minimizer s.

33.6 Stopping Parameters

A minimization procedure should stop when one of the following conditions is true:

• A minimum has been found to within the user-specified precision.

• A user-specified maximum number of iterations has been reached.

• An error has occurred.

The handling of these conditions is under user control. The function below allows the user to test the precision of the current result.

[Function] int gsl_min_test_interval (double x_lower, double x_upper,

double epsabs, double epsrel) This function tests for the convergence of the interval [x lower, x upper] with absolute error epsabs and relative error epsrel. The test returns GSL_SUCCESS if the following condition is achieved,

|a − b| < epsabs + epsrel min(|a|, |b|)

when the interval x = [a, b] does not include the origin. If the interval includes the origin then min(|a|, |b|) is replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error is accurately estimated for minima close to the origin.

This condition on the interval also implies that any estimate of the minimum x_{m} in the interval satisfies the same condition with respect to the true minimum x_{m}^{∗},

|x_{m} − x_{m}^{∗}| < epsabs + epsrel x_{m}^{∗}

assuming that the true minimum x_{m}^{∗} is contained within the interval.

33.7 Minimization Algorithms

The minimization algorithms described in this section require an initial interval which is guaranteed to contain a minimum—if a and b are the endpoints of the interval and x is an estimate of the minimum then f(a) > f(x) < f(b). This ensures that the function has at least one minimum somewhere in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.

[Minimizer] gsl_min_fminimizer_goldensection

The golden section algorithm is the simplest method of bracketing the minimum of a function. It is the slowest algorithm provided by the library, with linear convergence. On each iteration, the algorithm first compares the subintervals from the endpoints to the current minimum. The larger subinterval is divided in a golden section (using the famous ratio (3 − \sqrt{5})/2 = 0.3189660…) and the value of the function at this new point is calculated. The new value is used with the constraint f(a′) > f(x′) < f(b′) to a select new interval containing the minimum, by discarding the least useful point. This procedure can be continued indefinitely until the interval is sufficiently small. Choosing the golden section as the bisection ratio can be shown to provide the fastest convergence for this type of algorithm.

[Minimizer] gsl_min_fminimizer_brent

The Brent minimization algorithm combines a parabolic interpolation with the golden section algorithm. This produces a fast algorithm which is still robust. The outline of the algorithm can be summarized as follows: on each iteration Brent’s method approximates the function using an interpolating parabola through three existing points. The minimum of the parabola is taken as a guess for the minimum. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary golden section step. The full details of Brent’s method include some additional checks to improve convergence.

33.8 Examples

The following program uses the Brent algorithm to find the minimum of the function f(x) = cos(x) + 1, which occurs at x = π. The starting interval is (0, 6), with an initial guess for the minimum of 2.

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_min.h>
double fn1 (double x, void * params)
{
  return cos(x) + 1.0;
}
int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_min_fminimizer_type *T;
  gsl_min_fminimizer *s;
  double m = 2.0, m_expected = M_PI;
  double a = 0.0, b = 6.0;

gsl_function F;
F.function = &fn1;
F.params = 0;
T = gsl_min_fminimizer_brent;
s = gsl_min_fminimizer_alloc (T);
gsl_min_fminimizer_set (s, &F, m, a, b);
printf ("using %s method\n",
        gsl_min_fminimizer_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n",
        "iter", "lower", "upper", "min",
        "err", "err(est)");
printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
        iter, a, b,
        m, m - m_expected, b - a);
do
  {
    iter++;
    status = gsl_min_fminimizer_iterate (s);
    m = gsl_min_fminimizer_x_minimum (s);
    a = gsl_min_fminimizer_x_lower (s);
    b = gsl_min_fminimizer_x_upper (s);
    status
      = gsl_min_test_interval (a, b, 0.001, 0.0);
    if (status == GSL_SUCCESS)
      printf ("Converged:\n");
    printf ("%5d [%.7f, %.7f] "
            "%.7f %.7f %+.7f %.7f\n",
            iter, a, b,
            m, m_expected, m - m_expected, b - a);
  }
while (status == GSL_CONTINUE && iter < max_iter);
return status;
}

Here are the results of the minimization procedure.

$ ./a.out
  0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000

Chapter 33: One dimensional Minimization 340

1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
Converged:
11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175

33.9 References and Further Reading

Further information on Brent’s algorithm is available in the following book, Richard Brent, Algorithms for minimization without derivatives, Prentice-Hall (1973), republished by Dover in paperback (2002), ISBN 0-486-41998-3.

34 Multidimensional Root-Finding

This chapter describes functions for multidimensional root-finding (solving nonlinear systems with n equations in n unknowns). The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs. The solvers are based on the original Fortran library minpack.

The header file ‘gsl_multiroots.h’ contains prototypes for the multidimensional root finding functions and related declarations.

34.1 Overview

The problem of multidimensional root finding requires the simultaneous solution of n equations, f_{i}, in n variables, x_{i},

f_{i}(x_{1}, …, x_{n}) = 0 for i = 1 … n.

In general there are no bracketing methods available for n dimensional systems, and no way of knowing whether any solutions exist. All algorithms proceed from an initial guess using a variant of the Newton iteration,

x → x′ = x − J^{−1}f(x)

where x, f are vector quantities and J is the Jacobian matrix J_{ij} = ∂f_{i}/∂x_{j}. Additional strategies can be used to enlarge the region of convergence. These include requiring a decrease in the norm |f| on each step proposed by Newton’s method, or taking steepestdescent steps in the direction of the negative gradient of |f|.

Several root-finding algorithms are available within a single framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

• initialize solver state, s, for algorithm T

• update s using the iteration T

• test s for convergence, and repeat iteration if necessary

The evaluation of the Jacobian matrix can be problematic, either because programming the derivatives is intractable or because computation of the n^{2} terms of the matrix becomes too expensive. For these reasons the algorithms provided by the library are divided into two classes according to whether the derivatives are available or not.

The state for solvers with an analytic Jacobian matrix is held in a gsl_multiroot_ fdfsolver struct. The updating procedure requires both the function and its derivatives to be supplied by the user.

The state for solvers which do not use an analytic Jacobian matrix is held in a gsl_ multiroot_fsolver struct. The updating procedure uses only function evaluations (not derivatives). The algorithms estimate the matrix J or J^{−1} by approximate methods.

34.2 Initializing the Solver

The following functions initialize a multidimensional solver, either with or without derivatives. The solver itself depends only on the dimension of the problem and the algorithm and can be reused for different problems.

[Function] gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc (const

gsl multiroot fsolver type * T, size t n) This function returns a pointer to a newly allocated instance of a solver of type T for a system of n dimensions. For example, the following code creates an instance of a hybrid solver, to solve a 3-dimensional system of equations.

     const gsl_multiroot_fsolver_type * T
         = gsl_multiroot_fsolver_hybrid;
     gsl_multiroot_fsolver * s
         = gsl_multiroot_fsolver_alloc (T, 3);

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc

(const gsl multiroot fdfsolver type * T, size t n) This function returns a pointer to a newly allocated instance of a derivative solver of type T for a system of n dimensions. For example, the following code creates an instance of a Newton-Raphson solver, for a 2-dimensional system of equations.

     const gsl_multiroot_fdfsolver_type * T
         = gsl_multiroot_fdfsolver_newton;
     gsl_multiroot_fdfsolver * s =
         gsl_multiroot_fdfsolver_alloc (T, 2);

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] int gsl_multiroot_fsolver_set (gsl multiroot fsolver * s,

gsl multiroot function * f, gsl vector * x) This function sets, or resets, an existing solver s to use the function f and the initial guess x.

[Function] int gsl_multiroot_fdfsolver_set (gsl multiroot fdfsolver * s,

gsl multiroot function fdf * fdf, gsl vector * x) This function sets, or resets, an existing solver s to use the function and derivative fdf and the initial guess x.

[Function] void gsl_multiroot_fsolver_free (gsl multiroot fsolver * s)

[Function] void gsl_multiroot_fdfsolver_free (gsl multiroot fdfsolver * s)

These functions free all the memory associated with the solver s.

[Function] const char * gsl_multiroot_fsolver_name (const

gsl multiroot fsolver * s)

[Function] const char * gsl_multiroot_fdfsolver_name (const

gsl multiroot fdfsolver * s) These functions return a pointer to the name of the solver. For example,

     printf ("s is a '%s' solver\n",
             gsl_multiroot_fdfsolver_name (s));

would print something like s is a 'newton' solver.

34.3 Providing the function to solve

You must provide n functions of n variables for the root finders to operate on. In order to allow for general parameters the functions are defined by the following data types:

[Data Type] gsl_multiroot_function

This data type defines a general system of functions with parameters.

int (* f) (const gsl_vector * x, void * params, gsl_vector * f)

this function should store the vector result f(x, params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.

size_t n the dimension of the system, i.e. the number of components of the vectors x and f.

void * params

a pointer to the parameters of the function.

Here is an example using Powell’s test function,

f_{1}(x) = Ax_{0}x_{1} − 1, f_{2}(x) = exp(−x_{0}) + exp(−x_{1}) − (1 + 1/A)

with A = 10^{4}. The following code defines a gsl_multiroot_function system F which you could pass to a solver:

struct powell_params { double A; };
int
powell (gsl_vector * x, void * p, gsl_vector * f) {
   struct powell_params * params
     = *(struct powell_params *)p;
   const double A = (params->A);
   const double x0 = gsl_vector_get(x,0);
   const double x1 = gsl_vector_get(x,1);
   gsl_vector_set (f, 0, A * x0 * x1 - 1);
   gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)
                          - (1.0 + 1.0/A)));
   return GSL_SUCCESS
}
gsl_multiroot_function F;
struct powell_params params = { 10000.0 };
F.f = &powell;
F.n = 2;
F.params = &params;

[Data Type] gsl_multiroot_function_fdf

This data type defines a general system of functions with parameters and the corresponding Jacobian matrix of derivatives,

int (* f) (const gsl_vector * x, void * params, gsl_vector * f)

this function should store the vector result f(x, params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.

int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)

this function should store the n-by-n matrix result J_{ij} = ∂f_{i}(x, params)/∂x_{j} in J for argument x and parameters params, returning an appropriate error code if the function cannot be computed.

int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f,
gsl_matrix * J)

This function should set the values of the f and J as above, for arguments x and parameters params. This function provides an optimization of the separate functions for f(x) and J(x)—it is always faster to compute the function and its derivative at the same time.

size_t n the dimension of the system, i.e. the number of components of the vectors x and f.

void * params

a pointer to the parameters of the function.

The example of Powell’s test function defined above can be extended to include analytic derivatives using the following code,

int
powell_df (gsl_vector * x, void * p, gsl_matrix * J)
{
   struct powell_params * params
     = *(struct powell_params *)p;
   const double A = (params->A);
   const double x0 = gsl_vector_get(x,0);
   const double x1 = gsl_vector_get(x,1);
   gsl_matrix_set (J, 0, 0, A * x1);
   gsl_matrix_set (J, 0, 1, A * x0);
   gsl_matrix_set (J, 1, 0, -exp(-x0));
   gsl_matrix_set (J, 1, 1, -exp(-x1));
   return GSL_SUCCESS
}
int
powell_fdf (gsl_vector * x, void * p,
            gsl_matrix * f, gsl_matrix * J) {
   struct powell_params * params
     = *(struct powell_params *)p;
   const double A = (params->A);

const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
const double u0 = exp(-x0);
const double u1 = exp(-x1);
gsl_vector_set (f, 0, A * x0 * x1 - 1);
gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));
gsl_matrix_set (J, 0, 0, A * x1);
gsl_matrix_set (J, 0, 1, A * x0);
gsl_matrix_set (J, 1, 0, -u0);
gsl_matrix_set (J, 1, 1, -u1);
return GSL_SUCCESS
}
gsl_multiroot_function_fdf FDF;
FDF.f = &powell_f;
FDF.df = &powell_df;
FDF.fdf = &powell_fdf;
FDF.n = 2;
FDF.params = 0;

Note that the function powell_fdf is able to reuse existing terms from the function when calculating the Jacobian, thus saving time.

34.4 Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code.

[Function] int gsl_multiroot_fsolver_iterate (gsl multiroot fsolver * s)
[Function] int gsl_multiroot_fdfsolver_iterate (gsl multiroot fdfsolver * s)

These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned,

GSL_EBADFUNC

the iteration encountered a singular point where the function or its derivative evaluated to Inf or NaN.

GSL_ENOPROG

the iteration is not making any progress, preventing the algorithm from continuing.

The solver maintains a current best estimate of the root at all times. This information can be accessed with the following auxiliary functions,

[Function] gsl_vector * gsl_multiroot_fsolver_root (const

gsl multiroot fsolver * s)

[Function] gsl_vector * gsl_multiroot_fdfsolver_root (const

gsl multiroot fdfsolver * s) These functions return the current estimate of the root for the solver s.

[Function] gsl_vector * gsl_multiroot_fsolver_f (const gsl multiroot fsolver

* s)

[Function] gsl_vector * gsl_multiroot_fdfsolver_f (const

gsl multiroot fdfsolver * s) These functions return the function value f(x) at the current estimate of the root for the solver s.

[Function] gsl_vector * gsl_multiroot_fsolver_dx (const

gsl multiroot fsolver * s)

[Function] gsl_vector * gsl_multiroot_fdfsolver_dx (const

gsl multiroot fdfsolver * s) These functions return the last step dx taken by the solver s.

34.5 Search Stopping Parameters

A root finding procedure should stop when one of the following conditions is true: • A multidimensional root has been found to within the user-specified precision.

• A user-specified maximum number of iterations has been reached.

• An error has occurred.

The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result in several standard ways.

[Function] int gsl_multiroot_test_delta (const gsl vector * dx, const

gsl vector * x, double epsabs, double epsrel) This function tests for the convergence of the sequence by comparing the last step dx with the absolute error epsabs and relative error epsrel to the current position x. The test returns GSL_SUCCESS if the following condition is achieved,

|dx_{i}| < epsabs + epsrel |x_{i}|

for each component of x and returns GSL_CONTINUE otherwise.

[Function] int gsl_multiroot_test_residual (const gsl vector * f, double

   epsabs)

This function tests the residual value f against the absolute error bound epsabs. The test returns GSL_SUCCESS if the following condition is achieved,

∑_{i} |f_{i}| < epsabs

and returns GSL_CONTINUE otherwise. This criterion is suitable for situations where the precise location of the root, x, is unimportant provided a value can be found where the residual is small enough.

34.6 Algorithms using Derivatives

The root finding algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the root, but there is no absolute guarantee of convergence—the function must be suitable for this technique and the initial guess must be sufficiently close to the root for it to work. When the conditions are satisfied then convergence is quadratic.

[Derivative Solver] gsl_multiroot_fdfsolver_hybridsj

This is a modified version of Powell’s Hybrid method as implemented in the hybrj algorithm in minpack. Minpack was written by Jorge J. Moré, Burton S. Garbow and Kenneth E. Hillstrom. The Hybrid algorithm retains the fast convergence of Newton’s method but will also reduce the residual when Newton’s method is unreliable. The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a proposed new position x′ must satisfy the condition |D(x′ − x)| < δ, where D is a diagonal scaling matrix and δ is the size of the trust region. The components of D are computed internally, using the column norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the behavior of the algorithm for badly scaled functions. On each iteration the algorithm first determines the standard Newton step by solving the system Jdx = −f. If this step falls inside the trust region it is used as a trial step in the next stage. If not, the algorithm uses the linear combination of the Newton and gradient directions which is predicted to minimize the norm of the function while staying inside the trust region,

dx = −αJ^{−1}f(x) − β∇|f(x)|^{2}.

This combination of Newton and gradient directions is referred to as a dogleg step. The proposed step is now tested by evaluating the function at the resulting point, x′. If the step reduces the norm of the function sufficiently then it is accepted and size of the trust region is increased. If the proposed step fails to improve the solution then the size of the trust region is decreased and another trial step is computed. The speed of the algorithm is increased by computing the changes to the Jacobian approximately, using a rank-1 update. If two successive attempts fail to reduce the residual then the full Jacobian is recomputed. The algorithm also monitors the progress of the solution and returns an error if several steps fail to make any improvement,

GSL_ENOPROG

the iteration is not making any progress, preventing the algorithm from continuing.

GSL_ENOPROGJ

re-evaluations of the Jacobian indicate that the iteration is not making any progress, preventing the algorithm from continuing.

[Derivative Solver] gsl_multiroot_fdfsolver_hybridj

This algorithm is an unscaled version of hybridsj. The steps are controlled by a spherical trust region |x′ − x| < δ, instead of a generalized region. This can be useful if the generalized region estimated by hybridsj is inappropriate.

[Derivative Solver] gsl_multiroot_fdfsolver_newton

Newton’s Method is the standard root-polishing algorithm. The algorithm begins with an initial guess for the location of the solution. On each iteration a linear approximation to the function F is used to estimate the step which will zero all the components of the residual. The iteration is defined by the following sequence,

x → x′ = x − J^{−1}f(x)

where the Jacobian matrix J is computed from the derivative functions provided by f. The step dx is obtained by solving the linear system,

J dx = −f(x)

using LU decomposition.

[Derivative Solver] gsl_multiroot_fdfsolver_gnewton

This is a modified version of Newton’s method which attempts to improve global convergence by requiring every step to reduce the Euclidean norm of the residual, |f(x)|. If the Newton step leads to an increase in the norm then a reduced step of relative size,

t = (\sqrt{1 + 6r} − 1)/(3r)

is proposed, with r being the ratio of norms |f(x′)|^{2}/|f(x)|^{2}. This procedure is repeated until a suitable step size is found.

34.7 Algorithms without Derivatives

The algorithms described in this section do not require any derivative information to be supplied by the user. Any derivatives needed are approximated by finite differences. Note that if the finite-differencing step size chosen by these routines is inappropriate, an explicit user-supplied numerical derivative can always be used with the algorithms described in the previous section.

[Solver] gsl_multiroot_fsolver_hybrids

This is a version of the Hybrid algorithm which replaces calls to the Jacobian function by its finite difference approximation. The finite difference approximation is computed

using gsl_multiroots_fdjac with a relative step size of GSL_SQRT_DBL_EPSILON.
[Solver] gsl_multiroot_fsolver_hybrid

This is a finite difference version of the Hybrid algorithm without internal scaling.

[Solver] gsl_multiroot_fsolver_dnewton

The discrete Newton algorithm is the simplest method of solving a multidimensional system. It uses the Newton iteration

x → x − J^{−1}f(x)

where the Jacobian matrix J is approximated by taking finite differences of the function f. The approximation scheme used by this implementation is,

J_{ij} = (f_{i}(x + δ_{j}) − f_{i}(x))/δ_{j}

where δ_{j} is a step of size \sqrt{ϵ}|x_{j}| with ϵ being the machine precision (ϵ ≈ 2.22×10^{−16}). The order of convergence of Newton’s algorithm is quadratic, but the finite differences require n^{2} function evaluations on each iteration. The algorithm may become unstable if the finite differences are not a good approximation to the true derivatives.

[Solver] gsl_multiroot_fsolver_broyden

The Broyden algorithm is a version of the discrete Newton algorithm which attempts to avoids the expensive update of the Jacobian matrix on each iteration. The changes to the Jacobian are also approximated, using a rank-1 update,

J^{−1} → J^{−1} − (J^{−1}df − dx)dx^{T} J^{−1}/dx^{T} J^{−1}df

where the vectors dx and df are the changes in x and f. On the first iteration the inverse Jacobian is estimated using finite differences, as in the discrete Newton algorithm. This approximation gives a fast update but is unreliable if the changes are not small, and the estimate of the inverse Jacobian becomes worse as time passes. The algorithm has a tendency to become unstable unless it starts close to the root. The Jacobian is refreshed if this instability is detected (consult the source for details). This algorithm is included only for demonstration purposes, and is not recommended for serious use.

34.8 Examples

The multidimensional solvers are used in a similar way to the one-dimensional root finding algorithms. This first example demonstrates the hybrids scaled-hybrid algorithm, which does not require derivatives. The program solves the Rosenbrock system of equations,

f_{1}(x, y) = a(1 − x), f_{2}(x, y) = b(y − x^{2})

with a = 1, b = 10. The solution of this system lies at (x, y) = (1, 1) in a narrow valley.

The first stage of the program is to define the system of equations,

#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_multiroots.h>
struct rparams
  {
    double a;
    double b;
  };
int
rosenbrock_f (const gsl_vector * x, void *params,
              gsl_vector * f)
{
  double a = ((struct rparams *) params)->a;
  double b = ((struct rparams *) params)->b;

const double x0 = gsl_vector_get (x, 0);
const double x1 = gsl_vector_get (x, 1);
const double y0 = a * (1 - x0);
const double y1 = b * (x1 - x0 * x0);
gsl_vector_set (f, 0, y0);
gsl_vector_set (f, 1, y1);
return GSL_SUCCESS;
}

The main program begins by creating the function object f, with the arguments (x,y) and parameters (a,b). The solver s is initialized to use this function, with the hybrids method.

int
main (void)
{
const gsl_multiroot_fsolver_type *T;
gsl_multiroot_fsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function f = {&rosenbrock_f, n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fsolver_hybrids;
s = gsl_multiroot_fsolver_alloc (T, 2);
gsl_multiroot_fsolver_set (s, &f, x);
print_state (iter, s);
do
  {
    iter++;
    status = gsl_multiroot_fsolver_iterate (s);
    print_state (iter, s);

           if (status)   /* check if solver is stuck */
             break;
           status =
             gsl_multiroot_test_residual (s->f, 1e-7);
         }
       while (status == GSL_CONTINUE && iter < 1000);
       printf ("status = %s\n", gsl_strerror (status));
       gsl_multiroot_fsolver_free (s);
       gsl_vector_free (x);
       return 0;
     }

Note that it is important to check the return status of each solver step, in case the algorithm becomes stuck. If an error condition is detected, indicating that the algorithm cannot proceed, then the error can be reported to the user, a new starting point chosen or a different algorithm used.

The intermediate state of the solution is displayed by the following function. The solver state contains the vector s->x which is the current position, and the vector s->f with corresponding function values.

     int
     print_state (size_t iter, gsl_multiroot_fsolver * s)
     {
       printf ("iter = %3u x = % .3f % .3f "
               "f(x) = % .3e % .3e\n",
               iter,
               gsl_vector_get (s->x, 0),
               gsl_vector_get (s->x, 1),
               gsl_vector_get (s->f, 0),
               gsl_vector_get (s->f, 1));
     }

Here are the results of running the program. The algorithm is started at (−10, −5) far from the solution. Since the solution is hidden in a narrow valley the earliest steps follow the gradient of the function downhill, in an attempt to reduce the large value of the residual. Once the root has been approximately located, on iteration 8, the Newton behavior takes over and convergence is very rapid.

      iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
      iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
      iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
      iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
      iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
      iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
      iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
      iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
      iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
      iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00

iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
status = success

Note that the algorithm does not update the location on every iteration. Some iterations are used to adjust the trust-region parameter, after trying a step which was found to be divergent, or to recompute the Jacobian, when poor convergence behavior is detected.

The next example program adds derivative information, in order to accelerate the solution. There are two derivative functions rosenbrock_df and rosenbrock_fdf. The latter computes both the function and its derivative simultaneously. This allows the optimization of any common terms. For simplicity we substitute calls to the separate f and df functions at this point in the code below.

int
rosenbrock_df (const gsl_vector * x, void *params,
             gsl_matrix * J)
{
const double a = ((struct rparams *) params)->a;
const double b = ((struct rparams *) params)->b;
const double x0 = gsl_vector_get (x, 0);
const double df00 = -a;
const double df01 = 0;
const double df10 = -2 * b  * x0;
const double df11 = b;
gsl_matrix_set (J, 0, 0, df00);
gsl_matrix_set (J, 0, 1, df01);
gsl_matrix_set (J, 1, 0, df10);
gsl_matrix_set (J, 1, 1, df11);
return GSL_SUCCESS;
}
int
rosenbrock_fdf (const gsl_vector * x, void *params,
              gsl_vector * f, gsl_matrix * J)
{
rosenbrock_f (x, params, f);
rosenbrock_df (x, params, J);
return GSL_SUCCESS;
}

The main program now makes calls to the corresponding fdfsolver versions of the functions,

int
main (void)

{
const gsl_multiroot_fdfsolver_type *T;
gsl_multiroot_fdfsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function_fdf f = {&rosenbrock_f,
                                &rosenbrock_df,
                                &rosenbrock_fdf,
                                n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fdfsolver_gnewton;
s = gsl_multiroot_fdfsolver_alloc (T, n);
gsl_multiroot_fdfsolver_set (s, &f, x);
print_state (iter, s);
do
  {
    iter++;
    status = gsl_multiroot_fdfsolver_iterate (s);
    print_state (iter, s);
    if (status)
      break;
    status = gsl_multiroot_test_residual (s->f, 1e-7);
  }
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));
gsl_multiroot_fdfsolver_free (s);
gsl_vector_free (x);
return 0;
}

The addition of derivative information to the hybrids solver does not make any significant difference to its behavior, since it able to approximate the Jacobian numerically with sufficient accuracy. To illustrate the behavior of a different derivative solver we switch to gnewton. This is a traditional Newton solver with the constraint that it scales back its step if the full step would lead “uphill”. Here is the output for the gnewton algorithm,

      iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
      iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
      iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
      iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
      status = success

The convergence is much more rapid, but takes a wide excursion out to the point (−4.23, −65.3). This could cause the algorithm to go astray in a realistic application. The hybrid algorithm follows the downhill path to the solution more reliably.

34.9 References and Further Reading

The original version of the Hybrid method is described in the following articles by Powell, M.J.D. Powell, “A Hybrid Method for Nonlinear Equations” (Chap 6, p 87–114) and “A Fortran Subroutine for Solving systems of Nonlinear Algebraic Equations” (Chap 7, p 115–161), in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, editor. Gordon and Breach, 1970.

The following papers are also relevant to the algorithms described in this section, J.J. Moré, M.Y. Cosnard, “Numerical Solution of Nonlinear Equations”, ACM Transactions on Mathematical Software, Vol 5, No 1, (1979), p 64–85 C.G. Broyden, “A Class of Methods for Solving Nonlinear Simultaneous Equations”, Mathematics of Computation, Vol 19 (1965), p 577–593 J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained Optimization Software”, ACM Transactions on Mathematical Software, Vol 7, No 1 (1981), p 17–41

35 Multidimensional Minimization

This chapter describes routines for finding minima of arbitrary multidimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, while providing full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs. The minimization algorithms can be used to maximize a function by inverting its sign.

The header file ‘gsl_multimin.h’ contains prototypes for the minimization functions and related declarations.

35.1 Overview

The problem of multidimensional minimization requires finding a point x such that the scalar function,

f(x_{1}, …, x_{n})

takes a value which is lower than at any neighboring point. For smooth functions the gradient g = ∇f vanishes at the minimum. In general there are no bracketing methods available for the minimization of n-dimensional functions. The algorithms proceed from an initial guess using a search algorithm which attempts to move in a downhill direction.

Algorithms making use of the gradient of the function perform a one-dimensional line minimisation along this direction until the lowest point is found to a suitable tolerance. The search direction is then updated with local information from the function and its derivatives, and the whole process repeated until the true n-dimensional minimum is found.

The Nelder-Mead Simplex algorithm applies a different strategy. It maintains n + 1 trial parameter vectors as the vertices of a n-dimensional simplex. In each iteration step it tries to improve the worst vertex by a simple geometrical transformation until the size of the simplex falls below a given tolerance.

Both types of algorithms use a standard framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

• initialize minimizer state, s, for algorithm T

• update s using the iteration T

• test s for convergence, and repeat iteration if necessary

Each iteration step consists either of an improvement to the line-minimisation in the current direction or an update to the search direction itself. The state for the minimizers is held in

a gsl_multimin_fdfminimizer struct or a gsl_multimin_fminimizer struct.

35.2 Caveats

Note that the minimization algorithms can only search for one local minimum at a time. When there are several local minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a local minimum in an area where there is more than one.

It is also important to note that the minimization algorithms find local minima; there is no way to determine whether a minimum is a global minimum of the function in question.

35.3 Initializing the Multidimensional Minimizer

The following function initializes a multidimensional minimizer. The minimizer itself depends only on the dimension of the problem and the algorithm and can be reused for different problems.

[Function] gsl_multimin_fdfminimizer *
   gsl_multimin_fdfminimizer_alloc (const

gsl multimin fdfminimizer type * T, size t n)

[Function] gsl_multimin_fminimizer * gsl_multimin_fminimizer_alloc

(const gsl multimin fminimizer type * T, size t n) This function returns a pointer to a newly allocated instance of a minimizer of type T for an n-dimension function. If there is insufficient memory to create the minimizer then the function returns a null pointer and the error handler is invoked with an error

code of GSL_ENOMEM.

[Function] int gsl_multimin_fdfminimizer_set (gsl multimin fdfminimizer * s,

gsl multimin function fdf * fdf, const gsl vector * x, double step_size, double tol)

This function initializes the minimizer s to minimize the function fdf starting from the initial point x. The size of the first trial step is given by step size. The accuracy of the line minimization is specified by tol. The precise meaning of this parameter depends on the method used. Typically the line minimization is considered successful if the gradient of the function g is orthogonal to the current search direction p to a relative accuracy of tol, where p · g < tol|p||g|.

[Function] int gsl_multimin_fminimizer_set (gsl multimin fminimizer * s,

gsl multimin function * f, const gsl vector * x, const gsl vector * step_size) This function initializes the minimizer s to minimize the function f, starting from the initial point x. The size of the initial trial steps is given in vector step size. The precise meaning of this parameter depends on the method used.

[Function] void gsl_multimin_fdfminimizer_free (gsl multimin fdfminimizer *

   s)

[Function] void gsl_multimin_fminimizer_free (gsl multimin fminimizer * s)

This function frees all the memory associated with the minimizer s.

[Function] const char * gsl_multimin_fdfminimizer_name (const

gsl multimin fdfminimizer * s)

[Function] const char * gsl_multimin_fminimizer_name (const

gsl multimin fminimizer * s) This function returns a pointer to the name of the minimizer. For example,

     printf ("s is a '%s' minimizer\n",
             gsl_multimin_fdfminimizer_name (s));
would print something like s is a 'conjugate_pr' minimizer.

35.4 Providing a function to minimize

You must provide a parametric function of n variables for the minimizers to operate on. You may also need to provide a routine which calculates the gradient of the function and a third routine which calculates both the function value and the gradient together. In order to allow for general parameters the functions are defined by the following data types:

[Data Type] gsl_multimin_function_fdf

This data type defines a general function of n variables with parameters and the corresponding gradient vector of derivatives,

double (* f) (const gsl_vector * x, void * params)

this function should return the result f(x, params) for argument x and parameters params.

void (* df) (const gsl_vector * x, void * params, gsl_vector * g)

this function should store the n-dimensional gradient g_{i} = ∂f(x, params)/∂x_{i} in the vector g for argument x and parameters params, returning an appropriate error code if the function cannot be computed.

void (* fdf) (const gsl_vector * x, void * params, double * f, gsl_vector *
g)

This function should set the values of the f and g as above, for arguments x and parameters params. This function provides an optimization of the separate functions for f(x) and g(x)—it is always faster to compute the function and its derivative at the same time.

size_t n the dimension of the system, i.e. the number of components of the vectors x.

void * params

a pointer to the parameters of the function.

[Data Type] gsl_multimin_function

This data type defines a general function of n variables with parameters,

double (* f) (const gsl_vector * x, void * params)

this function should return the result f(x, params) for argument x and parameters params.

size_t n the dimension of the system, i.e. the number of components of the vectors x.

void * params

a pointer to the parameters of the function.

The following example function defines a simple paraboloid with two parameters,

/* Paraboloid centered on (dp[0],dp[1]) */
double
my_f (const gsl_vector *v, void *params)
{

  double x, y;
  double *dp = (double *)params;
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
  return 10.0 * (x - dp[0]) * (x - dp[0]) +
           20.0 * (y - dp[1]) * (y - dp[1]) + 30.0;
}
/* The gradient of f, df = (df/dx, df/dy). */
void
my_df (const gsl_vector *v, void *params,
       gsl_vector *df)
{
  double x, y;
  double *dp = (double *)params;
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
  gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
  gsl_vector_set(df, 1, 40.0 * (y - dp[1]));
}
/* Compute both f and df together. */
void
my_fdf (const gsl_vector *x, void *params,
        double *f, gsl_vector *df)
{
  *f = my_f(x, params);
  my_df(x, params, df);
}

The function can be initialized using the following code,

gsl_multimin_function_fdf my_func;
double p[2] = { 1.0, 2.0 }; /* center at (1,2) */
my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.n = 2;
my_func.params = (void *)p;

35.5 Iteration

The following function drives the iteration of each algorithm. The function performs one iteration to update the state of the minimizer. The same function works for all minimizers so that different methods can be substituted at runtime without modifications to the code.

[Function] int gsl_multimin_fdfminimizer_iterate

(gsl multimin fdfminimizer * s)

[Function] int gsl_multimin_fminimizer_iterate (gsl multimin fminimizer *
        s)

These functions perform a single iteration of the minimizer s. If the iteration encounters an unexpected problem then an error code will be returned.

The minimizer maintains a current best estimate of the minimum at all times. This information can be accessed with the following auxiliary functions,

[Function] gsl_vector * gsl_multimin_fdfminimizer_x (const

gsl multimin fdfminimizer * s)

[Function] gsl_vector * gsl_multimin_fminimizer_x (const

gsl multimin fminimizer * s)

[Function] double gsl_multimin_fdfminimizer_minimum (const

gsl multimin fdfminimizer * s)

[Function] double gsl_multimin_fminimizer_minimum (const

gsl multimin fminimizer * s)

[Function] gsl_vector * gsl_multimin_fdfminimizer_gradient (const

gsl multimin fdfminimizer * s)

[Function] double gsl_multimin_fminimizer_size (const

gsl multimin fminimizer * s) These functions return the current best estimate of the location of the minimum, the value of the function at that point, its gradient, and minimizer specific characteristic size for the minimizer s.

[Function] int gsl_multimin_fdfminimizer_restart

(gsl multimin fdfminimizer * s) This function resets the minimizer s to use the current point as a new starting point.

35.6 Stopping Criteria

A minimization procedure should stop when one of the following conditions is true: • A minimum has been found to within the user-specified precision.

• A user-specified maximum number of iterations has been reached.

• An error has occurred.

The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result.

[Function] int gsl_multimin_test_gradient (const gsl vector * g, double

        epsabs)

This function tests the norm of the gradient g against the absolute tolerance epsabs. The gradient of a multidimensional function goes to zero at a minimum. The test returns GSL_SUCCESS if the following condition is achieved,

|g| < epsabs

and returns GSL_CONTINUE otherwise. A suitable choice of epsabs can be made from the desired accuracy in the function for small variations in x. The relationship between these quantities is given by δf = g δx.

[Function] int gsl_multimin_test_size (const double size, double epsabs)

This function tests the minimizer specific characteristic size (if applicable to the used minimizer) against absolute tolerance epsabs. The test returns GSL_SUCCESS if the size is smaller than tolerance, otherwise GSL_CONTINUE is returned.

35.7 Algorithms

There are several minimization methods available. The best choice of algorithm depends on the problem. All of the algorithms use the value of the function and its gradient at each evaluation point, except for the Simplex algorithm which uses function values only.

[Minimizer] gsl_multimin_fdfminimizer_conjugate_fr

This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate gradient algorithm proceeds as a succession of line minimizations. The sequence of search directions is used to build up an approximation to the curvature of the function in the neighborhood of the minimum. An initial search direction p is chosen using the gradient, and line minimization is carried out in that direction. The accuracy of the line minimization is specified by the parameter tol. The minimum along this line occurs when the function gradient g and the search direction p are orthogonal. The line minimization terminates when p · g < tol|p||g|. The search direction is updated using the Fletcher-Reeves formula p′ = g′ − βg where β = −|g′|^{2}/|g|^{2}, and the line minimization is then repeated for the new search direction.

[Minimizer] gsl_multimin_fdfminimizer_conjugate_pr

This is the Polak-Ribiere conjugate gradient algorithm. It is similar to the Fletcher-Reeves method, differing only in the choice of the coefficient β. Both methods work well when the evaluation point is close enough to the minimum of the objective function that it is well approximated by a quadratic hypersurface.

[Minimizer] gsl_multimin_fdfminimizer_vector_bfgs

This is the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) conjugate gradient algorithm. It is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region.

[Minimizer] gsl_multimin_fdfminimizer_steepest_descent

The steepest descent algorithm follows the downhill gradient of the function at each step. When a downhill step is successful the step-size is increased by a factor of two. If the downhill step leads to a higher function value then the algorithm backtracks and the step size is decreased using the parameter tol. A suitable value of tol for most applications is 0.1. The steepest descent method is inefficient and is included only for demonstration purposes.

[Minimizer] gsl_multimin_fminimizer_nmsimplex

This is the Simplex algorithm of Nelder and Mead. It constructs n vectors p_{i} from the starting vector x and the vector step size as follows:

\begin{aligned}p_{0} & = (x_{0}, x_{1}, ⋯, x_{n}) \\ p_{1} & = (x_{0} + step size_{0}, x_{1}, ⋯, x_{n}) \\ p_{2} & = (x_{0}, x_{1} + step size_{1}, ⋯, x_{n}) \\ … & = … \\ p_{n} & = (x_{0}, x_{1}, ⋯, x_{n} + step size_{n}) \end{aligned}

These vectors form the n + 1 vertices of a simplex in n dimensions. On each iteration the algorithm tries to improve the parameter vector p_{i} corresponding to the highest function value by simple geometrical transformations. These are reflection, reflection followed by expansion, contraction and multiple contraction. Using these transformations the simplex moves through the parameter space towards the minimum, where it contracts itself. After each iteration, the best vertex is returned. Note, that due to the nature of the algorithm not every step improves the current best parameter vector. Usually several iterations are required. The routine calculates the minimizer specific characteristic size as the average distance from the geometrical center of the simplex to all its vertices. This size can be used as a stopping criteria, as the simplex contracts itself near the minimum. The size is

returned by the function gsl_multimin_fminimizer_size.

35.8 Examples

This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in x and y, and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter.

int
main (void)
{
  size_t iter = 0;
  int status;
  const gsl_multimin_fdfminimizer_type *T;
  gsl_multimin_fdfminimizer *s;
  /* Position of the minimum (1,2). */
  double par[2] = { 1.0, 2.0 };
  gsl_vector *x;
  gsl_multimin_function_fdf my_func;
  my_func.f = &my_f;

Chapter 35: Multidimensional Minimization 362

my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.n = 2;
my_func.params = &par;
/* Starting point, x = (5,7) */
x = gsl_vector_alloc (2);
gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
T = gsl_multimin_fdfminimizer_conjugate_fr;
s = gsl_multimin_fdfminimizer_alloc (T, 2);
gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);
do
  {
    iter++;
    status = gsl_multimin_fdfminimizer_iterate (s);
    if (status)
      break;
    status = gsl_multimin_test_gradient (s->gradient, 1e-3);
    if (status == GSL_SUCCESS)
      printf ("Minimum found at:\n");
    printf ("%5d %.5f %.5f %10.5f\n", iter,
            gsl_vector_get (s->x, 0),
            gsl_vector_get (s->x, 1),
            s->f);
  }
while (status == GSL_CONTINUE && iter < 100);
gsl_multimin_fdfminimizer_free (s);
gsl_vector_free (x);
return 0;
}

The initial step-size is chosen as 0.01, a conservative estimate in this case, and the line minimization parameter is set at 0.0001. The program terminates when the norm of the gradient has been reduced below 0.001. The output of the program is shown below,

       x       y         f
  1 4.99629 6.99072  687.84780
  2 4.98886 6.97215  683.55456
  3 4.97400 6.93501  675.01278
  4 4.94429 6.86073  658.10798
  5 4.88487 6.71217  625.01340
  6 4.76602 6.41506  561.68440
  7 4.52833 5.82083  446.46694
  8 4.05295 4.63238  261.79422
  9 3.10219 2.25548   75.49762

        10 2.85185 1.62963   67.03704
        11 2.19088 1.76182   45.31640
        12 0.86892 2.02622   30.18555
     Minimum found at:
        13 1.00000 2.00000   30.00000

Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting the successive points.

The conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic. Additional iterations would be needed for a more complicated function.

Here is another example using the Nelder-Mead Simplex algorithm to minimize the same example object function, as above.

      int
      main(void)
      {
        size_t np = 2;
        double par[2] = {1.0, 2.0};
        const gsl_multimin_fminimizer_type *T =
          gsl_multimin_fminimizer_nmsimplex;
        gsl_multimin_fminimizer *s = NULL;
        gsl_vector *ss, *x;
        gsl_multimin_function minex_func;
        size_t iter = 0, i;
        int status;
        double size;
        /* Initial vertex size vector */
        ss = gsl_vector_alloc (np);

Chapter 35: Multidimensional Minimization 364

/* Set all step sizes to 1 */
gsl_vector_set_all (ss, 1.0);
/* Starting point */
x = gsl_vector_alloc (np);
gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
/* Initialize method and iterate */
minex_func.f = &my_f;
minex_func.n = np;
minex_func.params = (void *)&par;
s = gsl_multimin_fminimizer_alloc (T, np);
gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
do
  {
    iter++;
    status = gsl_multimin_fminimizer_iterate(s);
    if (status)
      break;
    size = gsl_multimin_fminimizer_size (s);
    status = gsl_multimin_test_size (size, 1e-2);
    if (status == GSL_SUCCESS)
      {
        printf ("converged to minimum at\n");
      }
    printf ("%5d ", iter);
    for (i = 0; i < np; i++)
      {
        printf ("%10.3e ", gsl_vector_get (s->x, i));
      }
    printf ("f() = %7.3f size = %.3f\n", s->fval, size);
  }
while (status == GSL_CONTINUE && iter < 100);
gsl_vector_free(x);
gsl_vector_free(ss);
gsl_multimin_fminimizer_free (s);
return status;
}

The minimum search stops when the Simplex size drops to 0.01. The output is shown below.

  1  6.500e+00  5.000e+00 f() = 512.500 size = 1.082
  2  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
  3  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
  4  5.500e+00  1.000e+00 f() = 252.500 size = 1.372
  5  2.625e+00  3.500e+00 f() = 101.406 size = 1.823
  6  3.469e+00  1.375e+00 f() = 98.760  size = 1.526

 7  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
 8  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
 9  1.016e+00  2.812e+00 f() = 43.206  size = 1.105
10  2.041e+00  2.008e+00 f() = 40.838  size = 0.645
11  1.236e+00  1.664e+00 f() = 32.816  size = 0.645
12  1.236e+00  1.664e+00 f() = 32.816  size = 0.447
13  5.225e-01  1.980e+00 f() = 32.288  size = 0.447
14  1.103e+00  2.073e+00 f() = 30.214  size = 0.345
15  1.103e+00  2.073e+00 f() = 30.214  size = 0.264
16  1.103e+00  2.073e+00 f() = 30.214  size = 0.160
17  9.864e-01  1.934e+00 f() = 30.090  size = 0.132
18  9.190e-01  1.987e+00 f() = 30.069  size = 0.092
19  1.028e+00  2.017e+00 f() = 30.013  size = 0.056
20  1.028e+00  2.017e+00 f() = 30.013  size = 0.046
21  1.028e+00  2.017e+00 f() = 30.013  size = 0.033
22  9.874e-01  1.985e+00 f() = 30.006  size = 0.028
23  9.846e-01  1.995e+00 f() = 30.003  size = 0.023
24  1.007e+00  2.003e+00 f() = 30.001  size = 0.012
converged to minimum at
25  1.007e+00  2.003e+00 f() = 30.001  size = 0.010

The simplex size first increases, while the simplex moves towards the minimum. After a while the size begins to decrease as the simplex contracts around the minimum.

35.9 References and Further Reading

A brief description of multidimensional minimization algorithms and further references can be found in the following book,

C.W. Ueberhuber, Numerical Computation (Volume 2), Chapter 14, Section 4.4 “Minimization Methods”, p. 325–335, Springer (1997), ISBN 3-540-62057-5.

The simplex algorithm is described in the following paper, J.A. Nelder and R. Mead, A simplex method for function minimization, Computer Journal vol. 7 (1965), 308–315.

36 Least-Squares Fitting

This chapter describes routines for performing least squares fits to experimental data using linear combinations of functions. The data may be weighted or unweighted, i.e. with known or unknown errors. For weighted data the functions compute the best fit parameters and their associated covariance matrix. For unweighted data the covariance matrix is estimated from the scatter of the points, giving a variance-covariance matrix.

The functions are divided into separate versions for simple one- or two-parameter regression and multiple-parameter fits. The functions are declared in the header file ‘gsl_fit.h’.

36.1 Overview

Least-squares fits are found by minimizing χ^{2} (chi-squared), the weighted sum of squared residuals over n experimental datapoints (x_{i}, y_{i}) for the model Y (c, x),

χ^{2} = ∑_{i} w_{i}(y_{i} − Y (c, x_{i}))^{2}

The p parameters of the model are c = {c_{0}, c_{1}, …}. The weight factors w_{i} are given by w_{i} = 1/σ_{i}^{2}, where σ_{i} is the experimental error on the data-point y_{i}. The errors are assumed to be gaussian and uncorrelated. For unweighted data the chi-squared sum is computed without any weight factors.

The fitting routines return the best-fit parameters c and their p × p covariance matrix. The covariance matrix measures the statistical errors on the best-fit parameters resulting from the errors on the data σ_{i}, and is defined as C_{ab} = ⟨δc_{a}δc_{b}⟩ where ⟨ ⟩ denotes an average over the gaussian error distributions of the underlying datapoints.

The covariance matrix is calculated by error propagation from the data errors σ_{i}. The change in a fitted parameter δc_{a} caused by a small change in the data δy_{i} is given by

δc_{a} = ∑_{i} \frac{∂c_{a}}{∂y_{i}} δy_{i}

allowing the covariance matrix to be written in terms of the errors on the data,

C_{ab} = ∑_{i,j} \frac{∂c_{a}}{∂y_{i}} \frac{∂c_{b}}{∂y_{j}} ⟨δy_{i}δy_{j}⟩

For uncorrelated data the fluctuations of the underlying datapoints satisfy ⟨δy_{i}δy_{j}⟩ = σ_{i}^{2}δ_{ij}, giving a corresponding parameter covariance matrix of

C_{ab} = ∑_{i} \frac{1}{w_{i}} \frac{∂c_{a}}{∂y_{i}} \frac{∂c_{b}}{∂y_{i}}

When computing the covariance matrix for unweighted data, i.e. data with unknown errors, the weight factors w_{i} in this sum are replaced by the single estimate w = 1/σ^{2}, where σ^{2} is the computed variance of the residuals about the best-fit model, σ^{2} = ∑(y_{i}−Y (c, x_{i}))^{2}/(n− p). This is referred to as the variance-covariance matrix.

The standard deviations of the best-fit parameters are given by the square root of the corresponding diagonal elements of the covariance matrix, σ_{c_{a}} = \sqrt{C_{aa}}.

36.2 Linear regression

The functions described in this section can be used to perform least-squares fits to a straight line model, Y (c, x) = c_{0} + c_{1}x.

[Function] int gsl_fit_linear (const double * x, const size t xstride, const double * y, const size t ystride, size t n, double * c0, double * c1, double * cov00, double * cov01, double * cov11, double * sumsq)

This function computes the best-fit linear regression coefficients (c0,c1) of the model Y = c_{0} + c_{1}X for the dataset (x, y), two vectors of length n with strides xstride and ystride. The errors on y are assumed unknown so the variance-covariance matrix for the parameters (c0, c1) is estimated from the scatter of the points around the best-fit line and returned via the parameters (cov00, cov01, cov11). The sum of squares of the residuals from the best-fit line is returned in sumsq.

[Function] int gsl_fit_wlinear (const double * x, const size t xstride, const double * w, const size t wstride, const double * y, const size t ystride, size t n, double * c0, double * c1, double * cov00, double * cov01, double *

   cov11, double * chisq)

This function computes the best-fit linear regression coefficients (c0,c1) of the model Y = c_{0} + c_{1}X for the weighted dataset (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length n and stride wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in y.

The covariance matrix for the parameters (c0, c1) is computed using the weights and returned via the parameters (cov00, cov01, cov11). The weighted sum of squares of the residuals from the best-fit line, χ^{2}, is returned in chisq.

[Function] int gsl_fit_linear_est (double x, double c0, double c1, double c00,

double c01, double c11, double * y, double * y_err) This function uses the best-fit linear regression coefficients c0,c1 and their covariance cov00,cov01,cov11 to compute the fitted function y and its standard deviation y err for the model Y = c_{0} + c_{1}X at the point x.

36.3 Linear fitting without a constant term

The functions described in this section can be used to perform least-squares fits to a straight line model without a constant term, Y = c_{1}X.

[Function] int gsl_fit_mul (const double * x, const size t xstride, const double * y, const size t ystride, size t n, double * c1, double * cov11, double *

   sumsq)

This function computes the best-fit linear regression coefficient c1 of the model Y = c_{1}X for the datasets (x, y), two vectors of length n with strides xstride and ystride. The errors on y are assumed unknown so the variance of the parameter c1 is estimated from the scatter of the points around the best-fit line and returned via the parameter cov11. The sum of squares of the residuals from the best-fit line is returned in sumsq.

[Function] int gsl_fit_wmul (const double * x, const size t xstride, const double * w, const size t wstride, const double * y, const size t ystride, size t n, double * c1, double * cov11, double * sumsq)

This function computes the best-fit linear regression coefficient c1 of the model Y = c_{1}X for the weighted datasets (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length n and stride wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in y. The variance of the parameter c1 is computed using the weights and returned via the parameter cov11. The weighted sum of squares of the residuals from the best-fit line, χ^{2}, is returned in chisq.

[Function] int gsl_fit_mul_est (double x, double c1, double c11, double * y,

double * y_err) This function uses the best-fit linear regression coefficient c1 and its covariance cov11 to compute the fitted function y and its standard deviation y err for the model Y = c_{1}X at the point x.

36.4 Multi-parameter fitting

The functions described in this section perform least-squares fits to a general linear model, y = Xc where y is a vector of n observations, X is an n by p matrix of predictor variables, and the elements of the vector c are the p unknown best-fit parameters which are to be estimated.

This formulation can be used for fits to any number of functions and/or variables by preparing the n-by-p matrix X appropriately. For example, to fit to a p-th order polynomial in x, use the following matrix,

X_{ij} = x_{i}^{j}

where the index i runs over the observations and the index j runs from 0 to p − 1. To fit to a set of p sinusoidal functions with fixed frequencies ω_{1}, ω_{2}, …, ω_{p}, use,

X_{ij} = sin(ω_{j}x_{i})

To fit to p independent variables x_{1}, x_{2}, …, x_{p}, use,

X_{ij} = x_{j}(i)

where x_{j}(i) is the i-th value of the predictor variable x_{j}.

The functions described in this section are declared in the header file ‘gsl_multifit.h’. The solution of the general linear least-squares system requires an additional working

space for intermediate results, such as the singular value decomposition of the matrix X.

[Function] gsl_multifit_linear_workspace * gsl_multifit_linear_alloc

(size t n, size t p) This function allocates a workspace for fitting a model to n observations using p parameters.

[Function] void gsl_multifit_linear_free (gsl multifit linear workspace *

        work)

This function frees the memory associated with the workspace w.

[Function] int gsl_multifit_linear (const gsl matrix * X, const gsl vector * y, gsl vector * c, gsl matrix * cov, double * chisq, gsl multifit linear workspace

   * work)

[Function] int gsl_multifit_linear_svd (const gsl matrix * X, const gsl vector * y, double tol, size t * rank, gsl vector * c, gsl matrix * cov, double * chisq, gsl multifit linear workspace * work)

These functions compute the best-fit parameters c of the model y = Xc for the observations y and the matrix of predictor variables X. The variance-covariance matrix of the model parameters cov is estimated from the scatter of the observations about the best-fit. The sum of squares of the residuals from the best-fit, χ^{2}, is returned in chisq. The best-fit is found by singular value decomposition of the matrix X using the preallocated workspace provided in work. The modified Golub-Reinsch SVD algorithm is used, with column scaling to improve the accuracy of the singular values. Any components which have zero singular value (to machine precision) are discarded from the fit. In the second form of the function the components are discarded if the ratio of singular values s_{i}/s_{0} falls below the user-specified tolerance tol, and the effective rank is returned in rank.

[Function] int gsl_multifit_wlinear (const gsl matrix * X, const gsl vector * w, const gsl vector * y, gsl vector * c, gsl matrix * cov, double * chisq, gsl multifit linear workspace * work)

[Function] int gsl_multifit_wlinear_svd (const gsl matrix * X, const gsl vector * w, const gsl vector * y, double tol, size t * rank, gsl vector * c, gsl matrix * cov, double * chisq, gsl multifit linear workspace * work)

This function computes the best-fit parameters c of the weighted model y = Xc for the observations y with weights w and the matrix of predictor variables X. The covariance matrix of the model parameters cov is computed with the given weights. The weighted sum of squares of the residuals from the best-fit, χ^{2}, is returned in chisq. The best-fit is found by singular value decomposition of the matrix X using the preallocated workspace provided in work. Any components which have zero singular value (to machine precision) are discarded from the fit. In the second form of the function the components are discarded if the ratio of singular values s_{i}/s_{0} falls below the user-specified tolerance tol, and the effective rank is returned in rank.

[Function] int gsl_multifit_linear_est (const gsl vector * x, const gsl vector *

c, const gsl matrix * cov, double * y, double * y_err) This function uses the best-fit multilinear regression coefficients c and their covariance matrix cov to compute the fitted function value y and its standard deviation y err for the model y = x.c at the point x.

36.5 Examples

The following program computes a least squares straight-line fit to a simple dataset, and outputs the best-fit line and its associated one standard-deviation error bars.

#include <stdio.h>

#include <gsl/gsl_fit.h>
int
main (void)
{
int i, n = 4;
double x[4] = { 1970, 1980, 1990, 2000 };
double y[4] = {   12,   11,   14,   13 };
double w[4] = {  0.1,  0.2,  0.3,  0.4 };
double c0, c1, cov00, cov01, cov11, chisq;
gsl_fit_wlinear (x, 1, w, 1, y, 1, n,
                 &c0, &c1, &cov00, &cov01, &cov11,
                 &chisq);
printf ("# best fit: Y = %g + %g X\n", c0, c1);
printf ("# covariance matrix:\n");
printf ("# [ %g, %g\n#   %g, %g]\n",
        cov00, cov01, cov01, cov11);
printf ("# chisq = %g\n", chisq);
for (i = 0; i < n; i++)
  printf ("data: %g %g %g\n",
                 x[i], y[i], 1/sqrt(w[i]));
printf ("\n");
for (i = -30; i < 130; i++)
  {
    double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
    double yf, yf_err;
    gsl_fit_linear_est (xf,
                        c0, c1,
                        cov00, cov01, cov11,
                        &yf, &yf_err);
    printf ("fit: %g %g\n", xf, yf);
    printf ("hi : %g %g\n", xf, yf + yf_err);
    printf ("lo : %g %g\n", xf, yf - yf_err);
  }
return 0;
}

The following commands extract the data from the output of the program and display it using the gnu plotutils graph utility,

$ ./demo > tmp
$ more tmp
# best fit: Y = -106.6 + 0.06 X
# covariance matrix:
# [ 39602, -19.9
#   -19.9, 0.01]
# chisq = 0.8
$ for n in data fit hi lo ;
   do
     grep "^$n" tmp | cut -d: -f2 > $n ;
   done
$ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
     -S 0 -I a -m 1 fit -m 2 hi -m 2 lo

The next program performs a quadratic fit y = c_{0} + c_{1}x + c_{2}x^{2} to a weighted dataset using the generalised linear fitting function gsl_multifit_wlinear. The model matrix X for a quadratic fit is given by,

X = ( \begin{matrix}1 & x_{0} & x_{0}^{2} \\ 1 & x_{1} & x_{1}^{2} \\ 1 & x_{2} & x_{2}^{2} \\ … & … & … \end{matrix} )

where the column of ones corresponds to the constant term c_{0}. The two remaining columns corresponds to the terms c_{1}x and c_{2}x^{2}.

The program reads n lines of data in the format (x, y, err) where err is the error (standard deviation) in the value y.

#include <stdio.h>
#include <gsl/gsl_multifit.h>
int

main (int argc, char **argv)
{
int i, n;
double xi, yi, ei, chisq;
gsl_matrix *X, *cov;
gsl_vector *y, *w, *c;
if (argc != 2)
  {
    fprintf (stderr,"usage: fit n < data\n");
    exit (-1);
  }
n = atoi (argv[1]);
X = gsl_matrix_alloc (n, 3);
y = gsl_vector_alloc (n);
w = gsl_vector_alloc (n);
c = gsl_vector_alloc (3);
cov = gsl_matrix_alloc (3, 3);
for (i = 0; i < n; i++)
  {
    int count = fscanf (stdin, "%lg %lg %lg",
                        &xi, &yi, &ei);
    if (count != 3)
      {
        fprintf (stderr, "error reading file\n");
        exit (-1);
      }
    printf ("%g %g +/- %g\n", xi, yi, ei);
    gsl_matrix_set (X, i, 0, 1.0);
    gsl_matrix_set (X, i, 1, xi);
    gsl_matrix_set (X, i, 2, xi*xi);
    gsl_vector_set (y, i, yi);
    gsl_vector_set (w, i, 1.0/(ei*ei));
  }
{
  gsl_multifit_linear_workspace * work
    = gsl_multifit_linear_alloc (n, 3);
  gsl_multifit_wlinear (X, w, y, c, cov,

                        &chisq, work);
  gsl_multifit_linear_free (work);
}
#define C(i) (gsl_vector_get(c,(i)))
#define COV(i,j) (gsl_matrix_get(cov,(i),(j)))
{
  printf ("# best fit: Y = %g + %g X + %g X^2\n",
          C(0), C(1), C(2));
  printf ("# covariance matrix:\n");
  printf ("[ %+.5e, %+.5e, %+.5e  \n",
             COV(0,0), COV(0,1), COV(0,2));
  printf ("  %+.5e, %+.5e, %+.5e  \n",
             COV(1,0), COV(1,1), COV(1,2));
  printf ("  %+.5e, %+.5e, %+.5e ]\n",
             COV(2,0), COV(2,1), COV(2,2));
  printf ("# chisq = %g\n", chisq);
}
return 0;
}

A suitable set of data for fitting can be generated using the following program. It outputs a set of points with gaussian errors from the curve y = e^{x} in the region 0 < x < 2.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_randist.h>
int
main (void)
{
double x;
const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (x = 0.1; x < 2; x+= 0.1)
  {
    double y0 = exp (x);
    double sigma = 0.1 * y0;
    double dy = gsl_ran_gaussian (r, sigma);

      printf ("%g %g %g\n", x, y0 + dy, sigma);
    }
  return 0;
}

The data can be prepared by running the resulting executable program,

$ ./generate > exp.dat
$ more exp.dat
0.1 0.97935 0.110517
0.2 1.3359 0.12214
0.3 1.52573 0.134986
0.4 1.60318 0.149182
0.5 1.81731 0.164872
0.6 1.92475 0.182212
....

To fit the data use the previous program, with the number of data points given as the first argument. In this case there are 19 data points.

$ ./fit 19 < exp.dat
0.1 0.97935 +/- 0.110517
0.2 1.3359 +/- 0.12214
...
# best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
# covariance matrix:
[ +1.25612e-02, -3.64387e-02, +1.94389e-02
  -3.64387e-02, +1.42339e-01, -8.48761e-02
  +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
# chisq = 23.0987

The parameters of the quadratic fit match the coefficients of the expansion of e^{x}, taking into account the errors on the parameters and the O(x^{3}) difference between the exponential and quadratic functions for the larger values of x. The errors on the parameters are given by the square-root of the corresponding diagonal elements of the covariance matrix. The chi-squared per degree of freedom is 1.4, indicating a reasonable fit to the data.

36.6 References and Further Reading

A summary of formulas and techniques for least squares fitting can be found in the “Statistics” chapter of the Annual Review of Particle Physics prepared by the Particle Data Group,

Review of Particle Properties, R.M. Barnett et al., Physical Review D54, 1 (1996)

    http://pdg.lbl.gov/

The Review of Particle Physics is available online at the website given above.

The tests used to prepare these routines are based on the NIST Statistical Reference Datasets. The datasets and their documentation are available from NIST at the following website,

              http://www.nist.gov/itl/div898/strd/index.html.

37 Nonlinear Least-Squares Fitting

This chapter describes functions for multidimensional nonlinear least-squares fitting. The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs.

The header file ‘gsl_multifit_nlin.h’ contains prototypes for the multidimensional nonlinear fitting functions and related declarations.

37.1 Overview

The problem of multidimensional nonlinear least-squares fitting requires the minimization of the squared residuals of n functions, f_{i}, in p parameters, x_{i},

Φ(x) = \frac{1}{2}||F (x)||^{2} = \frac{1}{2} ∑_{i=1}^{n} f_{i}(x_{1}, …, x_{p})^{2}

All algorithms proceed from an initial guess using the linearization,

ψ(p) = ||F (x + p)|| ≈ ||F (x) + Jp ||

where x is the initial point, p is the proposed step and J is the Jacobian matrix J_{ij} = ∂f_{i}/∂x_{j}. Additional strategies are used to enlarge the region of convergence. These include requiring a decrease in the norm ||F || on each step or using a trust region to avoid steps which fall outside the linear regime.

To perform a weighted least-squares fit of a nonlinear model Y (x, t) to data (t_{i}, y_{i}) with independent gaussian errors σ_{i}, use function components of the following form,

f_{i} = \frac{(Y (x, t_{i}) − y_{i})}{σ_{i}}

Note that the model parameters are denoted by x in this chapter since the non-linear leastsquares algorithms are described geometrically (i.e. finding the minimum of a surface). The independent variable of any data to be fitted is denoted by t.

With the definition above the Jacobian is J_{ij} = (1/σ_{i})∂Y_{i}/∂x_{j}, where Y_{i} = Y (x, t_{i}).

37.2 Initializing the Solver

[Function] gsl_multifit_fsolver * gsl_multifit_fsolver_alloc (const

gsl multifit fsolver type * T, size t n, size t p) This function returns a pointer to a newly allocated instance of a solver of type T for n observations and p parameters. The number of observations n must be greater than or equal to parameters p. If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] gsl_multifit_fdfsolver * gsl_multifit_fdfsolver_alloc

(const gsl multifit fdfsolver type * T, size t n, size t p) This function returns a pointer to a newly allocated instance of a derivative solver of type T for n observations and p parameters. For example, the following code creates an instance of a Levenberg-Marquardt solver for 100 data points and 3 parameters,

     const gsl_multifit_fdfsolver_type * T
         = gsl_multifit_fdfsolver_lmder;
     gsl_multifit_fdfsolver * s
         = gsl_multifit_fdfsolver_alloc (T, 100, 3);

The number of observations n must be greater than or equal to parameters p. If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

[Function] int gsl_multifit_fsolver_set (gsl multifit fsolver * s,

gsl multifit function * f, gsl vector * x) This function initializes, or reinitializes, an existing solver s to use the function f and the initial guess x.

[Function] int gsl_multifit_fdfsolver_set (gsl multifit fdfsolver * s,

gsl function fdf * fdf, gsl vector * x) This function initializes, or reinitializes, an existing solver s to use the function and derivative fdf and the initial guess x.

[Function] void gsl_multifit_fsolver_free (gsl multifit fsolver * s)
[Function] void gsl_multifit_fdfsolver_free (gsl multifit fdfsolver * s)

These functions free all the memory associated with the solver s.

[Function] const char * gsl_multifit_fsolver_name (const

gsl multifit fdfsolver * s)

[Function] const char * gsl_multifit_fdfsolver_name (const

gsl multifit fdfsolver * s) These functions return a pointer to the name of the solver. For example,

     printf ("s is a '%s' solver\n",
             gsl_multifit_fdfsolver_name (s));

would print something like s is a 'lmder' solver.

37.3 Providing the Function to be Minimized

You must provide n functions of p variables for the minimization algorithms to operate on. In order to allow for arbitrary parameters the functions are defined by the following data types:

[Data Type] gsl_multifit_function

This data type defines a general system of functions with arbitrary parameters.

int (* f) (const gsl_vector * x, void * params, gsl_vector * f)

this function should store the vector result f(x, params) in f for argument x and arbitrary parameters params, returning an appropriate error code if the function cannot be computed. size_t n the number of functions, i.e. the number of components of the vector f.

size_t p the number of independent variables, i.e. the number of components of the vector x.

void * params

a pointer to the arbitrary parameters of the function.

[Data Type] gsl_multifit_function_fdf

This data type defines a general system of functions with arbitrary parameters and the corresponding Jacobian matrix of derivatives,

int (* f) (const gsl_vector * x, void * params, gsl_vector * f)

this function should store the vector result f(x, params) in f for argument x and arbitrary parameters params, returning an appropriate error code if the function cannot be computed.

int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)

this function should store the n-by-p matrix result J_{ij} = ∂f_{i}(x, params)/∂x_{j} in J for argument x and arbitrary parameters params, returning an appropriate error code if the function cannot be computed.

int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f,
gsl_matrix * J)

This function should set the values of the f and J as above, for arguments x and arbitrary parameters params. This function provides an optimization of the separate functions for f(x) and J(x)—it is always faster to compute the function and its derivative at the same time.

size_t n the number of functions, i.e. the number of components of the vector f.

size_t p the number of independent variables, i.e. the number of components of the vector x.

void * params

a pointer to the arbitrary parameters of the function.

Note that when fitting a non-linear model against experimental data, the data is passed to the functions above using the params argument and the trial best-fit parameters through the x argument.

37.4 Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code.

[Function] int gsl_multifit_fsolver_iterate (gsl multifit fsolver * s)
[Function] int gsl_multifit_fdfsolver_iterate (gsl multifit fdfsolver * s)

These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned. The solver maintains a current estimate of the best-fit parameters at all times.

The solver struct s contains the following entries, which can be used to track the progress of the solution:

gsl_vector * x

The current position.

gsl_vector * f

The function value at the current position.

gsl_vector * dx

The difference between the current position and the previous position, i.e. the last step, taken as a vector.

gsl_matrix * J

The Jacobian matrix at the current position (for the gsl_multifit_fdfsolver struct only)

The best-fit information also can be accessed with the following auxiliary functions,

[Function] gsl_vector * gsl_multifit_fsolver_position (const

gsl multifit fsolver * s)

[Function] gsl_vector * gsl_multifit_fdfsolver_position (const

gsl multifit fdfsolver * s) These functions return the current position (i.e. best-fit parameters) s->x of the solver s.

37.5 Search Stopping Parameters

A minimization procedure should stop when one of the following conditions is true: • A minimum has been found to within the user-specified precision.

• A user-specified maximum number of iterations has been reached.

• An error has occurred.

The handling of these conditions is under user control. The functions below allow the user to test the current estimate of the best-fit parameters in several standard ways.

[Function] int gsl_multifit_test_delta (const gsl vector * dx, const gsl vector

* x, double epsabs, double epsrel) This function tests for the convergence of the sequence by comparing the last step dx with the absolute error epsabs and relative error epsrel to the current position x. The test returns GSL_SUCCESS if the following condition is achieved,

|dx_{i}| < epsabs + epsrel |x_{i}|

for each component of x and returns GSL_CONTINUE otherwise.

[Function] int gsl_multifit_test_gradient (const gsl vector * g, double

        epsabs)

This function tests the residual gradient g against the absolute error bound epsabs. Mathematically, the gradient should be exactly zero at the minimum. The test returns GSL_SUCCESS if the following condition is achieved,

∑_{i} |g_{i}| < epsabs

and returns GSL_CONTINUE otherwise. This criterion is suitable for situations where the precise location of the minimum, x, is unimportant provided a value can be found where the gradient is small enough.

[Function] int gsl_multifit_gradient (const gsl matrix * J, const gsl vector *

f, gsl vector * g) This function computes the gradient g of Φ(x) = (1/2)||F (x)||^{2} from the Jacobian matrix J and the function values f, using the formula g = J^{T} f.

37.6 Minimization Algorithms using Derivatives

The minimization algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the minimum. There is no absolute guarantee of convergence—the function must be suitable for this technique and the initial guess must be sufficiently close to the minimum for it to work.

[Derivative Solver] gsl_multifit_fdfsolver_lmsder

This is a robust and efficient version of the Levenberg-Marquardt algorithm as implemented in the scaled lmder routine in minpack. Minpack was written by Jorge J. Moré, Burton S. Garbow and Kenneth E. Hillstrom. The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a proposed new position x′ must satisfy the condition |D(x′ − x)| < δ, where D is a diagonal scaling matrix and δ is the size of the trust region. The components of D are computed internally, using the column norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the behavior of the algorithm for badly scaled functions. On each iteration the algorithm attempts to minimize the linear system |F + Jp| subject to the constraint |Dp| < Δ. The solution to this constrained linear system is found using the Levenberg-Marquardt method. The proposed step is now tested by evaluating the function at the resulting point, x′. If the step reduces the norm of the function sufficiently, and follows the predicted behavior of the function within the trust region, then it is accepted and the size of the trust region is increased. If the proposed step fails to improve the solution, or differs significantly from the expected behavior within the trust region, then the size of the trust region is decreased and another trial step is computed. The algorithm also monitors the progress of the solution and returns an error if the changes in the solution are smaller than the machine precision. The possible error codes are,

GSL_ETOLF

the decrease in the function falls below machine precision

GSL_ETOLX

the change in the position vector falls below machine precision

GSL_ETOLG

the norm of the gradient, relative to the norm of the function, falls below machine precision These error codes indicate that further iterations will be unlikely to change the solution from its current value.

[Derivative Solver] gsl_multifit_fdfsolver_lmder

This is an unscaled version of the lmder algorithm. The elements of the diagonal scaling matrix D are set to 1. This algorithm may be useful in circumstances where the scaled version of lmder converges too slowly, or the function is already scaled appropriately.

37.7 Minimization Algorithms without Derivatives

There are no algorithms implemented in this section at the moment.

37.8 Computing the covariance matrix of best fit parameters

[Function] int gsl_multifit_covar (const gsl matrix * J, double epsrel,

gsl matrix * covar) This function uses the Jacobian matrix J to compute the covariance matrix of the best-fit parameters, covar. The parameter epsrel is used to remove linear-dependent columns when J is rank deficient. The covariance matrix is given by,

C = (J^{T} J)^{−1}

and is computed by QR decomposition of J with column-pivoting. Any columns of R which satisfy

|R_{kk}| ≤ epsrel|R_{11}|

are considered linearly-dependent and are excluded from the covariance matrix (the corresponding rows and columns of the covariance matrix are set to zero). If the minimisation uses the weighted least-squares function f_{i} = (Y (x, t_{i}) − y_{i})/σ_{i} then the covariance matrix above gives the statistical error on the best-fit parameters resulting from the gaussian errors σ_{i} on the underlying data y_{i}. This can be verified from the relation δf = Jδc and the fact that the fluctuations in f from the data y_{i} are normalised by σ_{i} and so satisfy ⟨δfδf^{T} ⟩ = I. For an unweighted least-squares function f_{i} = (Y (x, t_{i}) − y_{i}) the covariance matrix above should be multiplied by the variance of the residuals about the best-fit σ^{2} = ∑(y_{i} − Y (x, t_{i}))^{2}/(n − p) to give the variance-covariance matrix σ^{2}C. This estimates the statistical error on the best-fit parameters from the scatter of the underlying data. For more information about covariance matrices see Section 36.1 [Fitting Overview], page 366.

37.9 Examples

The following example program fits a weighted exponential model with background to experimental data, Y = A exp(−λt) + b. The first part of the program sets up the functions expb_f and expb_df to calculate the model and its Jacobian. The appropriate fitting function is given by,

f_{i} = ((A exp(−λt_{i}) + b) − y_{i})/σ_{i}

where we have chosen t_{i} = i. The Jacobian matrix J is the derivative of these functions with respect to the three parameters (A, λ, b). It is given by,

J_{ij} = \frac{∂f_{i}}{∂x_{j}}

where x_{0} = A, x_{1} = λ and x_{2} = b.

/* expfit.c -- model functions for exponential + background */
struct data {
size_t n;
double * y;
double * sigma;
};
int
expb_f (const gsl_vector * x, void *data,
      gsl_vector * f)
{
size_t n = ((struct data *)data)->n;
double *y = ((struct data *)data)->y;
double *sigma = ((struct data *) data)->sigma;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
double b = gsl_vector_get (x, 2);
size_t i;
for (i = 0; i < n; i++)
  {
    /* Model Yi = A * exp(-lambda * i) + b */
    double t = i;
    double Yi = A * exp (-lambda * t) + b;
    gsl_vector_set (f, i, (Yi - y[i])/sigma[i]);
  }
return GSL_SUCCESS;
}
int
expb_df (const gsl_vector * x, void *data,
       gsl_matrix * J)
{

  size_t n = ((struct data *)data)->n;
  double *sigma = ((struct data *) data)->sigma;
  double A = gsl_vector_get (x, 0);
  double lambda = gsl_vector_get (x, 1);
  size_t i;
  for (i = 0; i < n; i++)
    {
      /* Jacobian matrix J(i,j) = dfi / dxj, */
      /* where fi = (Yi - yi)/sigma[i],      */
      /*       Yi = A * exp(-lambda * i) + b  */
      /* and the xj are the parameters (A,lambda,b) */
      double t = i;
      double s = sigma[i];
      double e = exp(-lambda * t);
      gsl_matrix_set (J, i, 0, e/s);
      gsl_matrix_set (J, i, 1, -t * A * e/s);
      gsl_matrix_set (J, i, 2, 1/s);
    }
  return GSL_SUCCESS;
}
int
expb_fdf (const gsl_vector * x, void *data,
          gsl_vector * f, gsl_matrix * J)
{
  expb_f (x, data, f);
  expb_df (x, data, J);
  return GSL_SUCCESS;
}

The main part of the program sets up a Levenberg-Marquardt solver and some simulated random data. The data uses the known parameters (1.0,5.0,0.1) combined with gaussian noise (standard deviation = 0.1) over a range of 40 timesteps. The initial guess for the parameters is chosen as (0.0, 1.0, 0.0).

#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlin.h>
#include "expfit.c"

#define N 40
void print_state (size_t iter, gsl_multifit_fdfsolver * s);
int
main (void)
{
const gsl_multifit_fdfsolver_type *T;
gsl_multifit_fdfsolver *s;
int status;
size_t i, iter = 0;
const size_t n = N;
const size_t p = 3;
gsl_matrix *covar = gsl_matrix_alloc (p, p);
double y[N], sigma[N];
struct data d = { n, y, sigma};
gsl_multifit_function_fdf f;
double x_init[3] = { 1.0, 0.0, 0.0 };
gsl_vector_view x = gsl_vector_view_array (x_init, p);
const gsl_rng_type * type;
gsl_rng * r;
gsl_rng_env_setup();
type = gsl_rng_default;
r = gsl_rng_alloc (type);
f.f = &expb_f;
f.df = &expb_df;
f.fdf = &expb_fdf;
f.n = n;
f.p = p;
f.params = &d;
/* This is the data to be fitted */
for (i = 0; i < n; i++)
  {
    double t = i;
    y[i] = 1.0 + 5 * exp (-0.1 * t)
               + gsl_ran_gaussian (r, 0.1);
    sigma[i] = 0.1;
    printf ("data: %d %g %g\n", i, y[i], sigma[i]);

  };
T = gsl_multifit_fdfsolver_lmsder;
s = gsl_multifit_fdfsolver_alloc (T, n, p);
gsl_multifit_fdfsolver_set (s, &f, &x.vector);
print_state (iter, s);
do
  {
    iter++;
    status = gsl_multifit_fdfsolver_iterate (s);
    printf ("status = %s\n", gsl_strerror (status));
    print_state (iter, s);
    if (status)
      break;
    status = gsl_multifit_test_delta (s->dx, s->x,
                                      1e-4, 1e-4);
  }
while (status == GSL_CONTINUE && iter < 500);
gsl_multifit_covar (s->J, 0.0, covar);
#define FIT(i) gsl_vector_get(s->x, i)
#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
{
  double chi = gsl_blas_dnrm2(s->f);
  double dof = n - p;
  double c = GSL_MAX_DBL(1, chi / sqrt(dof));
  printf("chisq/dof = %g\n",  pow(chi, 2.0) / dof);
  printf ("A      = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
  printf ("lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
  printf ("b      = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
}
printf ("status = %s\n", gsl_strerror (status));
gsl_multifit_fdfsolver_free (s);
return 0;
}

     void
     print_state (size_t iter, gsl_multifit_fdfsolver * s)
     {
       printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f "
               "|f(x)| = %g\n",
               iter,
               gsl_vector_get (s->x, 0),
               gsl_vector_get (s->x, 1),
               gsl_vector_get (s->x, 2),
               gsl_blas_dnrm2 (s->f));
     }

The iteration terminates when the change in x is smaller than 0.0001, as both an absolute and relative change. Here are the results of running the program:

      iter: 0 x=1.00000000 0.00000000 0.00000000 |f(x)|=117.349
      status=success
      iter: 1 x=1.64659312 0.01814772 0.64659312 |f(x)|=76.4578
      status=success
      iter: 2 x=2.85876037 0.08092095 1.44796363 |f(x)|=37.6838
      status=success
      iter: 3 x=4.94899512 0.11942928 1.09457665 |f(x)|=9.58079
      status=success
      iter: 4 x=5.02175572 0.10287787 1.03388354 |f(x)|=5.63049
      status=success
      iter: 5 x=5.04520433 0.10405523 1.01941607 |f(x)|=5.44398
      status=success
      iter: 6 x=5.04535782 0.10404906 1.01924871 |f(x)|=5.44397
      chisq/dof = 0.800996
      A      = 5.04536 +/- 0.06028
      lambda = 0.10405 +/- 0.00316
      b      = 1.01925 +/- 0.03782
      status = success

The approximate values of the parameters are found correctly, and the chi-squared value indicates a good fit (the chi-squared per degree of freedom is approximately 1). In this case the errors on the parameters can be estimated from the square roots of the diagonal elements of the covariance matrix.

If the chi-squared value shows a poor fit (i.e. χ^{2}/(n − p) ≫ 1) then the error estimates obtained from the covariance matrix will be too small. In the example program the error estimates are multiplied by \sqrt{χ^{2}/(n − p)} in this case, a common way of increasing the errors for a poor fit. Note that a poor fit will result from the use an inappropriate model, and the scaled error estimates may then be outside the range of validity for gaussian errors.

37.10 References and Further Reading

The minpack algorithm is described in the following article, J.J. Moré, The Levenberg-Marquardt Algorithm: Implementation and Theory, Lecture Notes in Mathematics, v630 (1978), ed G. Watson.

The following paper is also relevant to the algorithms described in this section, J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained Optimization Software”, ACM Transactions on Mathematical Software, Vol 7, No 1 (1981), p 17–41.

38 Physical Constants

This chapter describes macros for the values of physical constants, such as the speed of light, c, and gravitational constant, G. The values are available in different unit systems, including the standard MKSA system (meters, kilograms, seconds, amperes) and the CGSM system (centimeters, grams, seconds, gauss), which is commonly used in Astronomy.

The definitions of constants in the MKSA system are available in the file ‘gsl_const_mksa.h’. The constants in the CGSM system are defined in ‘gsl_const_cgsm.h’. Dimensionless constants, such as the fine structure constant, which are pure numbers are defined in ‘gsl_const_num.h’.

The full list of constants is described briefly below. Consult the header files themselves for the values of the constants used in the library.

38.1 Fundamental Constants

GSL_CONST_MKSA_SPEED_OF_LIGHT

The speed of light in vacuum, c.

GSL_CONST_MKSA_VACUUM_PERMEABILITY

The permeability of free space, μ_{0}. This constant is defined in the MKSA system only.

GSL_CONST_MKSA_VACUUM_PERMITTIVITY

The permittivity of free space, ϵ_{0}. This constant is defined in the MKSA system only.

GSL_CONST_MKSA_PLANCKS_CONSTANT_H

Planck’s constant, h.

GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR

Planck’s constant divided by 2π, h̄.

GSL_CONST_NUM_AVOGADRO

Avogadro’s number, N_{a}.

GSL_CONST_MKSA_FARADAY

The molar charge of 1 Faraday.

GSL_CONST_MKSA_BOLTZMANN

The Boltzmann constant, k.

GSL_CONST_MKSA_MOLAR_GAS

The molar gas constant, R_{0}.

GSL_CONST_MKSA_STANDARD_GAS_VOLUME

The standard gas volume, V_{0}.

GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT

The Stefan-Boltzmann radiation constant, σ.

GSL_CONST_MKSA_GAUSS

The magnetic field of 1 Gauss.

38.2 Astronomy and Astrophysics

GSL_CONST_MKSA_ASTRONOMICAL_UNIT

The length of 1 astronomical unit (mean earth-sun distance), au.

GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT

The gravitational constant, G.

GSL_CONST_MKSA_LIGHT_YEAR

The distance of 1 light-year, ly.

GSL_CONST_MKSA_PARSEC

The distance of 1 parsec, pc.

GSL_CONST_MKSA_GRAV_ACCEL

The standard gravitational acceleration on Earth, g.

GSL_CONST_MKSA_SOLAR_MASS

The mass of the Sun.

38.3 Atomic and Nuclear Physics

GSL_CONST_MKSA_ELECTRON_CHARGE

The charge of the electron, e.

GSL_CONST_MKSA_ELECTRON_VOLT

The energy of 1 electron volt, eV.

GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS

The unified atomic mass, amu.

GSL_CONST_MKSA_MASS_ELECTRON

The mass of the electron, m_{e}.

GSL_CONST_MKSA_MASS_MUON

The mass of the muon, m_{μ}.

GSL_CONST_MKSA_MASS_PROTON

The mass of the proton, m_{p}.

GSL_CONST_MKSA_MASS_NEUTRON

The mass of the neutron, m_{n}.

GSL_CONST_NUM_FINE_STRUCTURE

The electromagnetic fine structure constant α.

GSL_CONST_MKSA_RYDBERG

The Rydberg constant, Ry, in units of energy. This is related to the Rydberg inverse wavelength R by Ry = hcR.

GSL_CONST_MKSA_BOHR_RADIUS

The Bohr radius, a_{0}.

GSL_CONST_MKSA_ANGSTROM

The length of 1 angstrom.

GSL_CONST_MKSA_BARN

The area of 1 barn.

GSL_CONST_MKSA_BOHR_MAGNETON

The Bohr Magneton, μ_{B}.

GSL_CONST_MKSA_NUCLEAR_MAGNETON

The Nuclear Magneton, μ_{N}.

GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT

The absolute value of the magnetic moment of the electron, μ_{e}. The physical magnetic moment of the electron is negative.

GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT

The magnetic moment of the proton, μ_{p}.

GSL_CONST_MKSA_THOMSON_CROSS_SECTION

The Thomson cross section, σ_{T}.

GSL_CONST_MKSA_DEBYE

The electric dipole moment of 1 Debye, D.

38.4 Measurement of Time

GSL_CONST_MKSA_MINUTE

The number of seconds in 1 minute.

GSL_CONST_MKSA_HOUR

The number of seconds in 1 hour.

GSL_CONST_MKSA_DAY

The number of seconds in 1 day.

GSL_CONST_MKSA_WEEK

The number of seconds in 1 week.

38.5 Imperial Units

GSL_CONST_MKSA_INCH

The length of 1 inch.

GSL_CONST_MKSA_FOOT

The length of 1 foot.

GSL_CONST_MKSA_YARD

The length of 1 yard.

GSL_CONST_MKSA_MILE

The length of 1 mile.

GSL_CONST_MKSA_MIL

The length of 1 mil (1/1000th of an inch).

38.6 Speed and Nautical Units

GSL_CONST_MKSA_KILOMETERS_PER_HOUR

The speed of 1 kilometer per hour.

GSL_CONST_MKSA_MILES_PER_HOUR

The speed of 1 mile per hour.

GSL_CONST_MKSA_NAUTICAL_MILE

The length of 1 nautical mile.

GSL_CONST_MKSA_FATHOM

The length of 1 fathom.

GSL_CONST_MKSA_KNOT

The speed of 1 knot.

38.7 Printers Units

GSL_CONST_MKSA_POINT

The length of 1 printer’s point (1/72 inch).

GSL_CONST_MKSA_TEXPOINT

The length of 1 TeX point (1/72.27 inch).

38.8 Volume, Area and Length

GSL_CONST_MKSA_MICRON

The length of 1 micron.

GSL_CONST_MKSA_HECTARE

The area of 1 hectare.

GSL_CONST_MKSA_ACRE

The area of 1 acre.

GSL_CONST_MKSA_LITER

The volume of 1 liter.

GSL_CONST_MKSA_US_GALLON

The volume of 1 US gallon.

GSL_CONST_MKSA_CANADIAN_GALLON

The volume of 1 Canadian gallon.

GSL_CONST_MKSA_UK_GALLON

The volume of 1 UK gallon.

GSL_CONST_MKSA_QUART

The volume of 1 quart.

GSL_CONST_MKSA_PINT

The volume of 1 pint.

38.9 Mass and Weight

GSL_CONST_MKSA_POUND_MASS

The mass of 1 pound.

GSL_CONST_MKSA_OUNCE_MASS

The mass of 1 ounce.

GSL_CONST_MKSA_TON

The mass of 1 ton.

GSL_CONST_MKSA_METRIC_TON

The mass of 1 metric ton (1000 kg).

GSL_CONST_MKSA_UK_TON

The mass of 1 UK ton.

GSL_CONST_MKSA_TROY_OUNCE

The mass of 1 troy ounce.

GSL_CONST_MKSA_CARAT

The mass of 1 carat.

GSL_CONST_MKSA_GRAM_FORCE

The force of 1 gram weight.

GSL_CONST_MKSA_POUND_FORCE

The force of 1 pound weight.

GSL_CONST_MKSA_KILOPOUND_FORCE

The force of 1 kilopound weight.

GSL_CONST_MKSA_POUNDAL

The force of 1 poundal.

38.10 Thermal Energy and Power

GSL_CONST_MKSA_CALORIE

The energy of 1 calorie.

GSL_CONST_MKSA_BTU

The energy of 1 British Thermal Unit, btu.

GSL_CONST_MKSA_THERM

The energy of 1 Therm.

GSL_CONST_MKSA_HORSEPOWER

The power of 1 horsepower.

38.11 Pressure

GSL_CONST_MKSA_BAR

The pressure of 1 bar.

GSL_CONST_MKSA_STD_ATMOSPHERE

The pressure of 1 standard atmosphere.

GSL_CONST_MKSA_TORR

The pressure of 1 torr.

GSL_CONST_MKSA_METER_OF_MERCURY

The pressure of 1 meter of mercury.

GSL_CONST_MKSA_INCH_OF_MERCURY

The pressure of 1 inch of mercury.

GSL_CONST_MKSA_INCH_OF_WATER

The pressure of 1 inch of water.

GSL_CONST_MKSA_PSI

The pressure of 1 pound per square inch.

38.12 Viscosity

GSL_CONST_MKSA_POISE

The dynamic viscosity of 1 poise.

GSL_CONST_MKSA_STOKES

The kinematic viscosity of 1 stokes.

38.13 Light and Illumination

GSL_CONST_MKSA_STILB

The luminance of 1 stilb.

GSL_CONST_MKSA_LUMEN

The luminous flux of 1 lumen.

GSL_CONST_MKSA_LUX

The illuminance of 1 lux.

GSL_CONST_MKSA_PHOT

The illuminance of 1 phot.

GSL_CONST_MKSA_FOOTCANDLE

The illuminance of 1 footcandle.

GSL_CONST_MKSA_LAMBERT

The luminance of 1 lambert.

GSL_CONST_MKSA_FOOTLAMBERT

The luminance of 1 footlambert.

38.14 Radioactivity

GSL_CONST_MKSA_CURIE

The activity of 1 curie.

GSL_CONST_MKSA_ROENTGEN

The exposure of 1 roentgen.

GSL_CONST_MKSA_RAD

The absorbed dose of 1 rad.

38.15 Force and Energy

GSL_CONST_MKSA_NEWTON

The SI unit of force, 1 Newton.

GSL_CONST_MKSA_DYNE

The force of 1 Dyne = 10^{−5} Newton.

GSL_CONST_MKSA_JOULE

The SI unit of energy, 1 Joule.

GSL_CONST_MKSA_ERG

The energy 1 erg = 10^{−7} Joule.

38.16 Prefixes

These constants are dimensionless scaling factors.

GSL_CONST_NUM_YOTTA

10^{24}

GSL_CONST_NUM_ZETTA

10^{21}

GSL_CONST_NUM_EXA

10^{18}

GSL_CONST_NUM_PETA

10^{15}

GSL_CONST_NUM_TERA

10^{12}

GSL_CONST_NUM_GIGA

10^{9}

GSL_CONST_NUM_MEGA

10^{6}

GSL_CONST_NUM_KILO

10^{3}

GSL_CONST_NUM_MILLI

10^{−3}

GSL_CONST_NUM_MICRO

10^{−6}

GSL_CONST_NUM_NANO

10^{−9}

GSL_CONST_NUM_PICO

10^{−12}

GSL_CONST_NUM_FEMTO

10^{−15}

GSL_CONST_NUM_ATTO

10^{−18}

GSL_CONST_NUM_ZEPTO

10^{−21}

GSL_CONST_NUM_YOCTO

10^{−24}

38.17 Examples

The following program demonstrates the use of the physical constants in a calculation. In this case, the goal is to calculate the range of light-travel times from Earth to Mars.

The required data is the average distance of each planet from the Sun in astronomical units (the eccentricities and inclinations of the orbits will be neglected for the purposes of this calculation). The average radius of the orbit of Mars is 1.52 astronomical units, and for the orbit of Earth it is 1 astronomical unit (by definition). These values are combined with the MKSA values of the constants for the speed of light and the length of an astronomical unit to produce a result for the shortest and longest light-travel times in seconds. The figures are converted into minutes before being displayed.

     #include <stdio.h>
     #include <gsl/gsl_const_mksa.h>
     int
     main (void)
     {
       double c  = GSL_CONST_MKSA_SPEED_OF_LIGHT;
       double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
       double minutes = GSL_CONST_MKSA_MINUTE;
       /* distance stored in meters */
       double r_earth = 1.00 * au;
       double r_mars  = 1.52 * au;
       double t_min, t_max;
       t_min = (r_mars - r_earth) / c;
       t_max = (r_mars + r_earth) / c;
       printf ("light travel time from Earth to Mars:\n");
       printf ("minimum = %.1f minutes\n", t_min / minutes);
       printf ("maximum = %.1f minutes\n", t_max / minutes);
       return 0;
     }

Here is the output from the program,

     light travel time from Earth to Mars:
     minimum = 4.3 minutes

     maximum = 21.0 minutes

38.18 References and Further Reading

The authoritative sources for physical constanst are the 2002 CODATA recommended values, published in the articles below. Further information on the values of physical constants is also available from the cited articles and the NIST website.

Journal of Physical and Chemical Reference Data, 28(6), 1713-1852, 1999 Reviews of Modern Physics, 72(2), 351-495, 2000

    http://www.physics.nist.gov/cuu/Constants/index.html
    http://physics.nist.gov/Pubs/SP811/appenB9.html

39 IEEE floating-point arithmetic

This chapter describes functions for examining the representation of floating point numbers and controlling the floating point environment of your program. The functions described in this chapter are declared in the header file ‘gsl_ieee_utils.h’.

39.1 Representation of floating point numbers

The IEEE Standard for Binary Floating-Point Arithmetic defines binary formats for single and double precision numbers. Each number is composed of three parts: a sign bit (s), an exponent (E) and a fraction (f). The numerical value of the combination (s, E, f) is given by the following formula,

(−1)^{s}(1 · fffff …)2^{E}

The sign bit is either zero or one. The exponent ranges from a minimum value E_{min} to a maximum value E_{max} depending on the precision. The exponent is converted to an unsigned number e, known as the biased exponent, for storage by adding a bias parameter, e = E + bias. The sequence fffff... represents the digits of the binary fraction f. The binary digits are stored in normalized form, by adjusting the exponent to give a leading digit of 1. Since the leading digit is always 1 for normalized numbers it is assumed implicitly and does not have to be stored. Numbers smaller than 2^{E_{min}} are be stored in denormalized form with a leading zero,

(−1)^{s}(0 · fffff …)2^{E_{min}}

This allows gradual underflow down to 2^{E_{min}−p} for p bits of precision. A zero is encoded with the special exponent of 2^{E_{min}−1} and infinities with the exponent of 2^{E_{max}+1}. The format for single precision numbers uses 32 bits divided in the following way,

      seeeeeeeefffffffffffffffffffffff
      s = sign bit, 1 bit
      e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
      f = fraction, 23 bits

The format for double precision numbers uses 64 bits divided in the following way,

      seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff
      s = sign bit, 1 bit
      e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
      f = fraction, 52 bits

It is often useful to be able to investigate the behavior of a calculation at the bit-level and the library provides functions for printing the IEEE representations in a human-readable form.

[Function] void gsl_ieee_fprintf_float (FILE * stream, const float * x)
[Function] void gsl_ieee_fprintf_double (FILE * stream, const double * x)

These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream stream. A pointer is used to pass the number indirectly, to avoid any undesired promotion from float to double. The output takes one of the following forms, NaN the Not-a-Number symbol

Inf, -Inf positive or negative infinity

1.fffff...*2^E, -1.fffff...*2^E

a normalized floating point number

0.fffff...*2^E, -0.fffff...*2^E

a denormalized floating point number

0, -0 positive or negative zero

The output can be used directly in GNU Emacs Calc mode by preceding it with 2# to indicate binary.

[Function] void gsl_ieee_printf_float (const float * x)
[Function] void gsl_ieee_printf_double (const double * x)

These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream stdout.

The following program demonstrates the use of the functions by printing the single and double precision representations of the fraction 1/3. For comparison the representation of the value promoted from single to double precision is also printed.

#include <stdio.h>
#include <gsl/gsl_ieee_utils.h>
int
main (void)
{
  float f = 1.0/3.0;
  double d = 1.0/3.0;
  double fd = f; /* promote from float to double */
  printf (" f="); gsl_ieee_printf_float(&f);
  printf ("\n");
  printf ("fd="); gsl_ieee_printf_double(&fd);
  printf ("\n");
  printf (" d="); gsl_ieee_printf_double(&d);
  printf ("\n");
  return 0;
}

The binary representation of 1/3 is 0.01010101.... The output below shows that the IEEE format normalizes this fraction to give a leading digit of 1,

 f= 1.01010101010101010101011*2^-2
fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
 d= 1.0101010101010101010101010101010101010101010101010101*2^-2

The output also shows that a single-precision number is promoted to double-precision by adding zeros in the binary representation.

39.2 Setting up your IEEE environment

The IEEE standard defines several modes for controlling the behavior of floating point operations. These modes specify the important properties of computer arithmetic: the direction used for rounding (e.g. whether numbers should be rounded up, down or to the nearest number), the rounding precision and how the program should handle arithmetic exceptions, such as division by zero.

Many of these features can now be controlled via standard functions such as fpsetround, which should be used whenever they are available. Unfortunately in the past there has been no universal API for controlling their behavior—each system has had its own low-level way of accessing them. To help you write portable programs GSL allows you to specify modes in a platform-independent way using the environment variable GSL_IEEE_MODE. The library then takes care of all the necessary machine-specific initializations for you when you call

the function gsl_ieee_env_setup.
[Function] void gsl_ieee_env_setup ()

This function reads the environment variable GSL_IEEE_MODE and attempts to set up the corresponding specified IEEE modes. The environment variable should be a list of keywords, separated by commas, like this,

GSL_IEEE_MODE = "keyword,keyword,..." where keyword is one of the following mode-names,

single-precision
double-precision
extended-precision
round-to-nearest
round-down
round-up
round-to-zero
mask-all
mask-invalid
mask-denormalized
mask-division-by-zero
mask-overflow
mask-underflow
trap-inexact
trap-common

If GSL_IEEE_MODE is empty or undefined then the function returns immediately and no attempt is made to change the system’s IEEE mode. When the modes from GSL_ IEEE_MODE are turned on the function prints a short message showing the new settings to remind you that the results of the program will be affected.

If the requested modes are not supported by the platform being used then the function calls the error handler and returns an error code of GSL_EUNSUP. When options are specified using this method, the resulting mode is based on a default setting of the highest available precision (double precision or extended precision, depending on the platform) in round-to-nearest mode, with all exceptions enabled apart from the inexact exception. The inexact exception is generated whenever rounding occurs, so it must generally be disabled in typical scientific calculations. All other floating-point exceptions are enabled by default, including underflows and the use of denormalized numbers, for safety. They can be disabled with the individual masksettings or together using mask-all. The following adjusted combination of modes is convenient for many purposes,

     GSL_IEEE_MODE="double-precision,"\
                     "mask-underflow,"\
                       "mask-denormalized"

This choice ignores any errors relating to small numbers (either denormalized, or underflowing to zero) but traps overflows, division by zero and invalid operations.

To demonstrate the effects of different rounding modes consider the following program which computes e, the base of natural logarithms, by summing a rapidly-decreasing series,

e = 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + … = 2.71828182846...

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_ieee_utils.h>
int
main (void)
{
  double x = 1, oldsum = 0, sum = 0;
  int i = 0;
  gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */
  do
    {
      i++;
      oldsum = sum;
      sum += x;
      x = x / i;
      printf ("i=%2d sum=%.18f error=%g\n",
              i, sum, sum - M_E);
      if (i > 30)
         break;

         }
       while (sum != oldsum);
       return 0;
     }

Here are the results of running the program in round-to-nearest mode. This is the IEEE default so it isn’t really necessary to specify it here,

     $ GSL_IEEE_MODE="round-to-nearest" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     i= 2 sum=2.000000000000000000 error=-0.718282
     ....
     i=18 sum=2.718281828459045535 error=4.44089e-16
     i=19 sum=2.718281828459045535 error=4.44089e-16

After nineteen terms the sum converges to within 4 × 10^{−16} of the correct value. If we now change the rounding mode to round-down the final result is less accurate,

     $ GSL_IEEE_MODE="round-down" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     ....
     i=19 sum=2.718281828459041094 error=-3.9968e-15

The result is about 4 × 10^{−15} below the correct value, an order of magnitude worse than the result obtained in the round-to-nearest mode.

If we change to rounding mode to round-up then the series no longer converges (the reason is that when we add each term to the sum the final result is always rounded up. This is guaranteed to increase the sum by at least one tick on each iteration). To avoid this problem we would need to use a safer converge criterion, such as while (fabs(sum - oldsum) > epsilon), with a suitably chosen value of epsilon.

Finally we can see the effect of computing the sum using single-precision rounding, in the default round-to-nearest mode. In this case the program thinks it is still using double precision numbers but the CPU rounds the result of each floating point operation to singleprecision accuracy. This simulates the effect of writing the program using single-precision float variables instead of double variables. The iteration stops after about half the number of iterations and the final result is much less accurate,

     $ GSL_IEEE_MODE="single-precision" ./a.out
     ....
     i=12 sum=2.718281984329223633 error=1.5587e-07

with an error of O(10^{−7}), which corresponds to single precision accuracy (about 1 part in 10^{7}). Continuing the iterations further does not decrease the error because all the subsequent results are rounded to the same value.

39.3 References and Further Reading

The reference for the IEEE standard is, ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic.

A more pedagogical introduction to the standard can be found in the following paper, David Goldberg: What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys, Vol. 23, No. 1 (March 1991), pages 5–48. Corrigendum: ACM Computing Surveys, Vol. 23, No. 3 (September 1991), page 413. and see also the sections by B. A. Wichmann and Charles B. Dunham in Surveyor’s Forum: “What Every Computer Scientist Should Know About Floating-Point Arithmetic”. ACM Computing Surveys, Vol. 24, No. 3 (September 1992), page 319.

A detailed textbook on IEEE arithmetic and its practical use is available from SIAM Press,

Michael L. Overton, Numerical Computing with IEEE Floating Point Arithmetic, SIAM Press, ISBN 0898715717.

Appendix A Debugging Numerical Programs

This chapter describes some tips and tricks for debugging numerical programs which use GSL.

A.1 Using gdb

Any errors reported by the library are passed to the function gsl_error. By running your programs under gdb and setting a breakpoint in this function you can automatically catch any library errors. You can add a breakpoint for every session by putting

     break gsl_error

into your ‘.gdbinit’ file in the directory where your program is started.

If the breakpoint catches an error then you can use a backtrace (bt) to see the call-tree, and the arguments which possibly caused the error. By moving up into the calling function you can investigate the values of variables at that point. Here is an example from the program fft/test_trap, which contains the following line,

      status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);

The function gsl_fft_complex_wavetable_alloc takes the length of an FFT as its first argument. When this line is executed an error will be generated because the length of an FFT is not allowed to be zero.

To debug this problem we start gdb, using the file ‘.gdbinit’ to define a breakpoint in

gsl_error,
      $ gdb test_trap
      GDB is free software and you are welcome to distribute copies
      of it under certain conditions; type "show copying" to see
      the conditions.  There is absolutely no warranty for GDB;
      type "show warranty" for details.  GDB 4.16 (i586-debian-linux),
      Copyright 1996 Free Software Foundation, Inc.
      Breakpoint 1 at 0x8050b1e: file error.c, line 14.

When we run the program this breakpoint catches the error and shows the reason for it.

      (gdb) run
      Starting program: test_trap
      Breakpoint 1, gsl_error (reason=0x8052b0d
          "length n must be positive integer",
          file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
          at error.c:14
      14        if (gsl_error_handler)

The first argument of gsl_error is always a string describing the error. Now we can look at the backtrace to see what caused the problem,

      (gdb) bt
      #0  gsl_error (reason=0x8052b0d
          "length n must be positive integer",
          file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
          at error.c:14
      #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
          wavetable=0xbffff778) at c_init.c:108
      #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)

Appendix A: Debugging Numerical Programs 404

    at test_trap.c:94
#3  0x80488be in ___crt_dummy__()

We can see that the error was generated in the function gsl_fft_complex_wavetable_ alloc when it was called with an argument of n=0. The original call came from line 94 in

the file ‘test_trap.c’.

By moving up to the level of the original call we can find the line that caused the error,

(gdb) up
#1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
    wavetable=0xbffff778) at c_init.c:108
108   GSL_ERROR ("length n must be positive integer", GSL_EDOM);
(gdb) up
#2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
    at test_trap.c:94
94    status = gsl_fft_complex_wavetable_alloc (0,
        &complex_wavetable);

Thus we have found the line that caused the problem. From this point we could also print out the values of other variables such as complex_wavetable.

A.2 Examining floating point registers

The contents of floating point registers can be examined using the command info float (on supported platforms).

(gdb) info float
     st0: 0xc4018b895aa17a945000  Valid Normal -7.838871e+308
     st1: 0x3ff9ea3f50e4d7275000  Valid Normal 0.0285946
     st2: 0x3fe790c64ce27dad4800  Valid Normal 6.7415931e-08
     st3: 0x3ffaa3ef0df6607d7800  Spec  Normal 0.0400229
     st4: 0x3c028000000000000000  Valid Normal 4.4501477e-308
     st5: 0x3ffef5412c22219d9000  Zero  Normal 0.9580257
     st6: 0x3fff8000000000000000  Valid Normal 1
     st7: 0xc4028b65a1f6d243c800  Valid Normal -1.566206e+309
   fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
   fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
    ftag: 0x3fff
     fip: 0x08048b5c
     fcs: 0x051a0023
  fopoff: 0x08086820
  fopsel: 0x002b

Individual registers can be examined using the variables $reg, where reg is the register name.

(gdb) p $st1
$1 = 0.02859464454261210347719

A.3 Handling floating point exceptions

It is possible to stop the program whenever a SIGFPE floating point exception occurs. This can be useful for finding the cause of an unexpected infinity or NaN. The current handler settings can be shown with the command info signal SIGFPE.

(gdb) info signal SIGFPE
Signal  Stop  Print  Pass to program Description
SIGFPE  Yes   Yes    Yes             Arithmetic exception

Unless the program uses a signal handler the default setting should be changed so that SIGFPE is not passed to the program, as this would cause it to exit. The command handle

SIGFPE stop nopass prevents this.
      (gdb) handle SIGFPE stop nopass
      Signal  Stop  Print  Pass to program Description
      SIGFPE  Yes   Yes    No              Arithmetic exception

Depending on the platform it may be necessary to instruct the kernel to generate signals for floating point exceptions. For programs using GSL this can be achieved using the GSL_ IEEE_MODE environment variable in conjunction with the function gsl_ieee_env_setup() as described in see Chapter 39 [IEEE floating-point arithmetic], page 397.

     (gdb) set env GSL_IEEE_MODE=double-precision

A.4 GCC warning options for numerical programs

Writing reliable numerical programs in C requires great care. The following GCC warning options are recommended when compiling numerical programs:

     gcc -ansi -pedantic -Werror -Wall -W
       -Wmissing-prototypes -Wstrict-prototypes
       -Wtraditional -Wconversion -Wshadow
       -Wpointer-arith -Wcast-qual -Wcast-align
       -Wwrite-strings -Wnested-externs
       -fshort-enums -fno-common -Dinline= -g -O4

For details of each option consult the manual Using and Porting GCC. The following table gives a brief explanation of what types of errors these options catch.

-ansi -pedantic

Use ANSI C, and reject any non-ANSI extensions. These flags help in writing portable programs that will compile on other systems.

-Werror Consider warnings to be errors, so that compilation stops. This prevents warnings from scrolling off the top of the screen and being lost. You won’t be able to compile the program until it is completely warning-free.

-Wall This turns on a set of warnings for common programming problems. You need -Wall, but it is not enough on its own.

-O2 Turn on optimization. The warnings for uninitialized variables in -Wall rely on the optimizer to analyze the code. If there is no optimization then these warnings aren’t generated.

-W This turns on some extra warnings not included in -Wall, such as missing return values and comparisons between signed and unsigned integers.

-Wmissing-prototypes -Wstrict-prototypes

Warn if there are any missing or inconsistent prototypes. Without prototypes it is harder to detect problems with incorrect arguments.

-Wtraditional

This warns about certain constructs that behave differently in traditional and ANSI C. Whether the traditional or ANSI interpretation is used might be unpredictable on other compilers.

-Wconversion

The main use of this option is to warn about conversions from signed to unsigned integers. For example, unsigned int x = -1. If you need to perform such a conversion you can use an explicit cast.

-Wshadow This warns whenever a local variable shadows another local variable. If two variables have the same name then it is a potential source of confusion.

-Wpointer-arith -Wcast-qual -Wcast-align

These options warn if you try to do pointer arithmetic for types which don’t have a size, such as void, if you remove a const cast from a pointer, or if you cast a pointer to a type which has a different size, causing an invalid alignment.

-Wwrite-strings

This option gives string constants a const qualifier so that it will be a compiletime error to attempt to overwrite them.

-fshort-enums

This option makes the type of enum as short as possible. Normally this makes an enum different from an int. Consequently any attempts to assign a pointerto-int to a pointer-to-enum will generate a cast-alignment warning.

-fno-common

This option prevents global variables being simultaneously defined in different object files (you get an error at link time). Such a variable should be defined in one file and referred to in other files with an extern declaration.

-Wnested-externs

This warns if an extern declaration is encountered within a function.

-Dinline=

The inline keyword is not part of ANSI C. Thus if you want to use -ansi with a program which uses inline functions you can use this preprocessor definition to remove the inline keywords.

-g It always makes sense to put debugging symbols in the executable so that you can debug it using gdb. The only effect of debugging symbols is to increase the size of the file, and you can use the strip command to remove them later if necessary.

A.5 References and Further Reading

The following books are essential reading for anyone writing and debugging numerical programs with gcc and gdb.

R.M. Stallman, Using and Porting GNU CC, Free Software Foundation, ISBN 1882114388 R.M. Stallman, R.H. Pesch, Debugging with GDB: The GNU Source-Level Debugger, Free Software Foundation, ISBN 1882114779

For a tutorial introduction to the GNU C Compiler and related programs, see B.J. Gough, An Introduction to GCC, Network Theory Ltd, ISBN 0954161793

Appendix B Contributors to GSL

(See the AUTHORS file in the distribution for up-to-date information.)

Mark Galassi Conceived GSL (with James Theiler) and wrote the design document. Wrote the simulated annealing package and the relevant chapter in the manual.

James Theiler Conceived GSL (with Mark Galassi). Wrote the random number generators and the relevant chapter in this manual.

Jim Davies Wrote the statistical routines and the relevant chapter in this manual.

Brian Gough FFTs, numerical integration, random number generators and distributions, root finding, minimization and fitting, polynomial solvers, complex numbers, physical constants, permutations, vector and matrix functions, histograms, statistics, ieee-utils, revised cblas Level 2 & 3, matrix decompositions, eigensystems, cumulative distribution functions, testing, documentation and releases.

Reid Priedhorsky Wrote and documented the initial version of the root finding routines while at Los Alamos National Laboratory, Mathematical Modeling and Analysis Group.

Gerard Jungman Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra, Eigensystems, Hankel Transforms.

Mike Booth Wrote the Monte Carlo library.

Jorma Olavi Tähtinen Wrote the initial complex arithmetic functions.

Thomas Walter Wrote the initial heapsort routines and cholesky decomposition.

Fabrice Rossi Multidimensional minimization.

Carlo Perassi Implementation of the random number generators in Knuth’s Seminumerical Algorithms, 3rd Ed.

Szymon Jaroszewicz Wrote the routines for generating combinations.

Nicolas Darnis Wrote the initial routines for canonical permutations.

Jason H. Stover Wrote the major cumulative distribution functions.

Ivo Alxneit Wrote the routines for wavelet transforms.

Tuomo Keskitalo Improved the implementation of the ODE solvers.

Thanks to Nigel Lowry for help in proofreading the manual.

Appendix C Autoconf Macros

For applications using autoconf the standard macro AC_CHECK_LIB can be used to link with GSL automatically from a configure script. The library itself depends on the presence of a cblas and math library as well, so these must also be located before linking with the main libgsl file. The following commands should be placed in the ‘configure.ac’ file to perform these tests,

     AC_CHECK_LIB(m,main)
     AC_CHECK_LIB(gslcblas,main)
     AC_CHECK_LIB(gsl,main)

It is important to check for libm and libgslcblas before libgsl, otherwise the tests will fail. Assuming the libraries are found the output during the configure stage looks like this,

     checking for main in -lm... yes
     checking for main in -lgslcblas... yes
     checking for main in -lgsl... yes

If the library is found then the tests will define the macros HAVE_LIBGSL, HAVE_

LIBGSLCBLAS, HAVE_LIBM and add the options -lgsl -lgslcblas -lm to the variable
LIBS.

The tests above will find any version of the library. They are suitable for general use, where the versions of the functions are not important. An alternative macro is available in the file ‘gsl.m4’ to test for a specific version of the library. To use this macro simply add the following line to your ‘configure.in’ file instead of the tests above:

     AM_PATH_GSL(GSL_VERSION,
                [action-if-found],
                [action-if-not-found])

The argument GSL_VERSION should be the two or three digit major.minor or major.minor.micro version number of the release you require. A suitable choice for

action-if-not-found is,
     AC_MSG_ERROR(could not find required version of GSL)

Then you can add the variables GSL_LIBS and GSL_CFLAGS to your Makefile.am files to obtain the correct compiler flags. GSL_LIBS is equal to the output of the gsl-config -- libs command and GSL_CFLAGS is equal to gsl-config --cflags command. For example,

     libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas

Note that the macro AM_PATH_GSL needs to use the C compiler so it should appear in the ‘configure.in’ file before the macro AC_LANG_CPLUSPLUS for programs that use C++.

To test for inline the following test should be placed in your ‘configure.in’ file,

     AC_C_INLINE
     if test "$ac_cv_c_inline" != no ; then
       AC_DEFINE(HAVE_INLINE,1)
       AC_SUBST(HAVE_INLINE)
     fi

and the macro will then be defined in the compilation flags or by including the file ‘config.h’ before any library headers.

The following autoconf test will check for extern inline,

      dnl Check for "extern inline", using a modified version
      dnl of the test for AC_C_INLINE from acspecific.mt
      dnl
      AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
      [ac_cv_c_extern_inline=no
      AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
      extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
      double foo (double x) { return x + 1.0; };],
      [  foo(1.0)  ],
      [ac_cv_c_extern_inline="yes"])
      ])
      if test "$ac_cv_c_extern_inline" != no ; then
        AC_DEFINE(HAVE_INLINE,1)
        AC_SUBST(HAVE_INLINE)
      fi

The substitution of portability functions can be made automatically if you use autoconf. For example, to test whether the BSD function hypot is available you can include the following line in the configure file ‘configure.in’ for your application,

     AC_CHECK_FUNCS(hypot)

and place the following macro definitions in the file ‘config.h.in’,

     /* Substitute gsl_hypot for missing system hypot */

     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

The application source files can then use the include command #include <config.h> to substitute gsl_hypot for each occurrence of hypot when hypot is not available.

Appendix D GSL CBLAS Library

The prototypes for the low-level cblas functions are declared in the file gsl_cblas.h. For the definition of the functions consult the documentation available from Netlib (see Section 12.3 [BLAS References and Further Reading], page 121).

D.1 Level 1

[Function] float cblas_sdsdot (const int N, const float alpha, const float * x, const int incx, const float * y, const int incy)

[Function] double cblas_dsdot (const int N, const float * x, const int incx, const float * y, const int incy)

[Function] float cblas_sdot (const int N, const float * x, const int incx, const float * y, const int incy)

[Function] double cblas_ddot (const int N, const double * x, const int incx, const double * y, const int incy)

[Function] void cblas_cdotu_sub (const int N, const void * x, const int incx, const void * y, const int incy, void * dotu)

[Function] void cblas_cdotc_sub (const int N, const void * x, const int incx, const void * y, const int incy, void * dotc)

[Function] void cblas_zdotu_sub (const int N, const void * x, const int incx, const void * y, const int incy, void * dotu)

[Function] void cblas_zdotc_sub (const int N, const void * x, const int incx, const void * y, const int incy, void * dotc)

[Function] float cblas_snrm2 (const int N, const float * x, const int incx)

[Function] float cblas_sasum (const int N, const float * x, const int incx)

[Function] double cblas_dnrm2 (const int N, const double * x, const int incx)

[Function] double cblas_dasum (const int N, const double * x, const int incx)

[Function] float cblas_scnrm2 (const int N, const void * x, const int incx)

[Function] float cblas_scasum (const int N, const void * x, const int incx)

[Function] double cblas_dznrm2 (const int N, const void * x, const int incx)

[Function] double cblas_dzasum (const int N, const void * x, const int incx)

[Function] CBLAS_INDEX cblas_isamax (const int N, const float * x, const int

        incx)

[Function] CBLAS_INDEX cblas_idamax (const int N, const double * x, const int

        incx)

[Function] CBLAS_INDEX cblas_icamax (const int N, const void * x, const int incx)

[Function] CBLAS_INDEX cblas_izamax (const int N, const void * x, const int incx)

[Function] void cblas_sswap (const int N, float * x, const int incx, float * y, const

        int incy)

[Function] void cblas_scopy (const int N, const float * x, const int incx, float * y, const int incy)

[Function] void cblas_saxpy (const int N, const float alpha, const float * x, const int incx, float * y, const int incy)

[Function] void cblas_dswap (const int N, double * x, const int incx, double * y, const int incy)

[Function] void cblas_dcopy (const int N, const double * x, const int incx, double * y, const int incy)

[Function] void cblas_daxpy (const int N, const double alpha, const double * x, const int incx, double * y, const int incy)

[Function] void cblas_cswap (const int N, void * x, const int incx, void * y, const

        int incy)

[Function] void cblas_ccopy (const int N, const void * x, const int incx, void * y, const int incy)

[Function] void cblas_caxpy (const int N, const void * alpha, const void * x, const int incx, void * y, const int incy)

[Function] void cblas_zswap (const int N, void * x, const int incx, void * y, const

        int incy)

[Function] void cblas_zcopy (const int N, const void * x, const int incx, void * y, const int incy)

[Function] void cblas_zaxpy (const int N, const void * alpha, const void * x, const int incx, void * y, const int incy)

[Function] void cblas_srotg (float * a, float * b, float * c, float * s)

[Function] void cblas_srotmg (float * d1, float * d2, float * b1, const float b2, float * P)

[Function] void cblas_srot (const int N, float * x, const int incx, float * y, const int incy, const float c, const float s)

[Function] void cblas_srotm (const int N, float * x, const int incx, float * y, const int incy, const float * P)

[Function] void cblas_drotg (double * a, double * b, double * c, double * s)

[Function] void cblas_drotmg (double * d1, double * d2, double * b1, const double b2, double * P)

[Function] void cblas_drot (const int N, double * x, const int incx, double * y, const int incy, const double c, const double s)

[Function] void cblas_drotm (const int N, double * x, const int incx, double * y, const int incy, const double * P)

[Function] void cblas_sscal (const int N, const float alpha, float * x, const int

incx)

[Function] void cblas_dscal (const int N, const double alpha, double * x, const

int incx)

[Function] void cblas_cscal (const int N, const void * alpha, void * x, const int

incx)

[Function] void cblas_zscal (const int N, const void * alpha, void * x, const int

incx)

[Function] void cblas_csscal (const int N, const float alpha, void * x, const int

incx)

[Function] void cblas_zdscal (const int N, const double alpha, void * x, const int

incx)

D.2 Level 2

[Function] void cblas_sgemv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const float alpha, const float * A, const int lda, const float * x, const int incx, const float beta, float * y, const int incy)

[Function] void cblas_sgbmv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const float alpha, const float * A, const int lda, const float * x, const int incx, const float beta, float * y, const int incy)

[Function] void cblas_strmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const float * A, const int lda, float * x, const

int incx)

[Function] void cblas_stbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const float * A, const int lda, float * x, const int incx)

[Function] void cblas_stpmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const float * Ap, float * x, const int incx)

[Function] void cblas_strsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const float * A, const int lda, float * x, const

int incx)

[Function] void cblas_stbsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const float * A, const int lda, float * x, const int incx)

[Function] void cblas_stpsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const float * Ap, float * x, const int incx)

[Function] void cblas_dgemv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const double alpha, const double * A, const int lda, const double * x, const int incx, const double beta, double * y, const int incy)

[Function] void cblas_dgbmv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const double alpha, const double * A, const int lda, const double * x, const int incx, const double beta, double * y, const int incy)

[Function] void cblas_dtrmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const double * A, const int lda, double * x, const int incx)

[Function] void cblas_dtbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const double * A, const int lda, double * x, const int incx)

[Function] void cblas_dtpmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const double * Ap, double * x, const int

incx)

[Function] void cblas_dtrsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const double * A, const int lda, double * x, const int incx)

[Function] void cblas_dtbsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const double * A, const int lda, double * x, const int incx)

[Function] void cblas_dtpsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const double * Ap, double * x, const int

incx)

[Function] void cblas_cgemv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_cgbmv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_ctrmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * A, const int lda, void * x, const

int incx)

[Function] void cblas_ctbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const void * A, const int lda, void * x, const int incx)

[Function] void cblas_ctpmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * Ap, void * x, const int incx)

[Function] void cblas_ctrsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * A, const int lda, void * x, const

int incx)

[Function] void cblas_ctbsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const void * A, const int lda, void * x, const int incx)

[Function] void cblas_ctpsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * Ap, void * x, const int incx)

[Function] void cblas_zgemv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_zgbmv (const enum CBLAS ORDER order, const enum CBLAS TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_ztrmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * A, const int lda, void * x, const

int incx)

[Function] void cblas_ztbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const void * A, const int lda, void * x, const int incx)

[Function] void cblas_ztpmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * Ap, void * x, const int incx)

[Function] void cblas_ztrsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * A, const int lda, void * x, const

int incx)

[Function] void cblas_ztbsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const int K, const void * A, const int lda, void * x, const int incx)

[Function] void cblas_ztpsv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int N, const void * Ap, void * x, const int incx)

[Function] void cblas_ssymv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const float * A, const int lda, const float * x, const int incx, const float beta, float * y, const int incy)

[Function] void cblas_ssbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const int K, const float alpha, const float * A, const int lda, const float * x, const int incx, const float beta, float * y, const int incy)

[Function] void cblas_sspmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const float * Ap, const float * x, const int incx, const float beta, float * y, const int incy)

[Function] void cblas_sger (const enum CBLAS ORDER order, const int M, const int N, const float alpha, const float * x, const int incx, const float * y, const int incy, float * A, const int lda)

[Function] void cblas_ssyr (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const float * x, const int incx, float * A, const int lda)

[Function] void cblas_sspr (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const float * x, const int incx, float * Ap)

[Function] void cblas_ssyr2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const float * x, const int incx, const float * y, const int incy, float * A, const int lda)

[Function] void cblas_sspr2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const float * x, const int incx, const float * y, const int incy, float * A)

[Function] void cblas_dsymv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const double * A, const int lda, const double * x, const int incx, const double beta, double * y, const

int incy)

[Function] void cblas_dsbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const int K, const double alpha, const double * A, const int lda, const double * x, const int incx, const double beta, double * y, const int incy)

[Function] void cblas_dspmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const double * Ap, const double * x, const int incx, const double beta, double * y, const int incy)

[Function] void cblas_dger (const enum CBLAS ORDER order, const int M, const int N, const double alpha, const double * x, const int incx, const double * y, const int incy, double * A, const int lda)

[Function] void cblas_dsyr (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const double * x, const int incx, double * A, const int lda)

[Function] void cblas_dspr (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const double * x, const int incx, double * Ap)

[Function] void cblas_dsyr2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const double * x, const int incx, const double * y, const int incy, double * A, const int lda)

[Function] void cblas_dspr2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const double * x, const int incx, const double * y, const int incy, double * A)

[Function] void cblas_chemv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int

incy)

[Function] void cblas_chbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const int K, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_chpmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * Ap, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_cgeru (const enum CBLAS ORDER order, const int M, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * A, const int lda)

[Function] void cblas_cgerc (const enum CBLAS ORDER order, const int M, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * A, const int lda)

[Function] void cblas_cher (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const void * x, const int incx, void * A, const int lda)

[Function] void cblas_chpr (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const float alpha, const void * x, const int incx, void * A)

[Function] void cblas_cher2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * A, const int lda)

[Function] void cblas_chpr2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * Ap)

[Function] void cblas_zhemv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int

incy)

[Function] void cblas_zhbmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const int K, const void * alpha, const void * A, const int lda, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_zhpmv (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * Ap, const void * x, const int incx, const void * beta, void * y, const int incy)

[Function] void cblas_zgeru (const enum CBLAS ORDER order, const int M, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * A, const int lda)

[Function] void cblas_zgerc (const enum CBLAS ORDER order, const int M, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * A, const int lda)

[Function] void cblas_zher (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const void * x, const int incx, void * A, const int lda)

[Function] void cblas_zhpr (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const double alpha, const void * x, const int incx, void * A)

[Function] void cblas_zher2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * A, const int lda)

[Function] void cblas_zhpr2 (const enum CBLAS ORDER order, const enum CBLAS UPLO Uplo, const int N, const void * alpha, const void * x, const int incx, const void * y, const int incy, void * Ap)

D.3 Level 3

[Function] void cblas_sgemm (const enum CBLAS ORDER Order, const enum CBLAS TRANSPOSE TransA, const enum CBLAS TRANSPOSE TransB, const int M, const int N, const int K, const float alpha, const float * A, const int lda, const float * B, const int ldb, const float beta, float * C, const int ldc)

[Function] void cblas_ssymm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const int M, const int N, const float alpha, const float * A, const int lda, const float * B, const int ldb, const float beta, float * C, const int ldc)

[Function] void cblas_ssyrk (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const float alpha, const float * A, const int lda, const float beta, float * C, const int ldc)

[Function] void cblas_ssyr2k (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const float alpha, const float * A, const int lda, const float * B, const int ldb, const float beta, float * C, const int ldc)

[Function] void cblas_strmm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const float alpha, const float * A, const int lda, float * B, const int

ldb)

[Function] void cblas_strsm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const float alpha, const float * A, const int lda, float * B, const int

ldb)

[Function] void cblas_dgemm (const enum CBLAS ORDER Order, const enum CBLAS TRANSPOSE TransA, const enum CBLAS TRANSPOSE TransB, const int M, const int N, const int K, const double alpha, const double * A, const int lda, const double * B, const int ldb, const double beta, double * C, const int ldc)

[Function] void cblas_dsymm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const int M, const int N, const double alpha, const double * A, const int lda, const double * B, const int ldb, const double beta, double * C, const int ldc)

[Function] void cblas_dsyrk (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const double alpha, const double * A, const int lda, const double beta, double * C, const int ldc)

[Function] void cblas_dsyr2k (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const double alpha, const double * A, const int lda, const double * B, const int ldb, const double beta, double * C, const int ldc)

[Function] void cblas_dtrmm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const double alpha, const double * A, const int lda, double * B, const int ldb)

[Function] void cblas_dtrsm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const double alpha, const double * A, const int lda, double * B, const int ldb)

[Function] void cblas_cgemm (const enum CBLAS ORDER Order, const enum CBLAS TRANSPOSE TransA, const enum CBLAS TRANSPOSE TransB, const int M, const int N, const int K, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_csymm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const int M, const int N, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_csyrk (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const void * alpha, const void * A, const int lda, const void * beta, void * C, const int ldc)

[Function] void cblas_csyr2k (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_ctrmm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const void * alpha, const void * A, const int lda, void * B, const int ldb)

[Function] void cblas_ctrsm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const void * alpha, const void * A, const int lda, void * B, const int ldb)

[Function] void cblas_zgemm (const enum CBLAS ORDER Order, const enum CBLAS TRANSPOSE TransA, const enum CBLAS TRANSPOSE TransB, const int M, const int N, const int K, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_zsymm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const int M, const int N, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_zsyrk (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const void * alpha, const void * A, const int lda, const void * beta, void * C, const int ldc)

[Function] void cblas_zsyr2k (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_ztrmm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const void * alpha, const void * A, const int lda, void * B, const int ldb)

[Function] void cblas_ztrsm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE TransA, const enum CBLAS DIAG Diag, const int M, const int N, const void * alpha, const void * A, const int lda, void * B, const int ldb)

[Function] void cblas_chemm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const int M, const int N, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_cherk (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const float alpha, const void * A, const int lda, const float beta, void * C, const int ldc)

[Function] void cblas_cher2k (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const float beta, void * C, const int ldc)

[Function] void cblas_zhemm (const enum CBLAS ORDER Order, const enum CBLAS SIDE Side, const enum CBLAS UPLO Uplo, const int M, const int N, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const void * beta, void * C, const int ldc)

[Function] void cblas_zherk (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const double alpha, const void * A, const int lda, const double beta, void * C, const int ldc)

[Function] void cblas_zher2k (const enum CBLAS ORDER Order, const enum CBLAS UPLO Uplo, const enum CBLAS TRANSPOSE Trans, const int N, const int K, const void * alpha, const void * A, const int lda, const void * B, const int ldb, const double beta, void * C, const int ldc)

[Function] void cblas_xerbla (int p, const char * rout, const char * form, ...)

D.4 Examples

The following program computes the product of two matrices using the Level-3 blas function sgemm,

( \begin{matrix}0.11 & 0.12 & 0.13 \\ 0.21 & 0.22 & 0.23 \end{matrix} ) ( \begin{matrix}1011 & 1012 \\ 1021 & 1022 \\ 1031 & 1031 \end{matrix} ) = ( \begin{matrix}367.76 & 368.12 \\ 674.06 & 674.72 \end{matrix} )

The matrices are stored in row major order but could be stored in column major order if the first argument of the call to cblas_sgemm was changed to CblasColMajor.

#include <stdio.h>
#include <gsl/gsl_cblas.h>
int
main (void)
{
int lda = 3;
float A[] = { 0.11, 0.12, 0.13,
              0.21, 0.22, 0.23 };
int ldb = 2;
float B[] = { 1011, 1012,
              1021, 1022,
              1031, 1032 };
int ldc = 2;
float C[] = { 0.00, 0.00,
              0.00, 0.00 };
/* Compute C = A B */
cblas_sgemm (CblasRowMajor,
             CblasNoTrans, CblasNoTrans, 2, 2, 3,
             1.0, A, lda, B, ldb, 0.0, C, ldc);
printf ("[ %g, %g\n", C[0], C[1]);
printf ("  %g, %g ]\n", C[2], C[3]);

       return 0;
     }

To compile the program use the following command line,

     $ gcc -Wall demo.c -lgslcblas

There is no need to link with the main library -lgsl in this case as the cblas library is an independent unit. Here is the output from the program,

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

Free Software Needs Free Documentation

The following article was written by Richard Stallman, founder of the GNU Project.

The biggest deficiency in the free software community today is not in the software—it is the lack of good free documentation that we can include with the free software. Many of our most important programs do not come with free reference manuals and free introductory texts. Documentation is an essential part of any software package; when an important free software package does not come with a free manual and a free tutorial, that is a major gap. We have many such gaps today.

Consider Perl, for instance. The tutorial manuals that people normally use are non-free. How did this come about? Because the authors of those manuals published them with restrictive terms—no copying, no modification, source files not available—which exclude them from the free software world.

That wasn’t the first time this sort of thing happened, and it was far from the last. Many times we have heard a GNU user eagerly describe a manual that he is writing, his intended contribution to the community, only to learn that he had ruined everything by signing a publication contract to make it non-free.

Free documentation, like free software, is a matter of freedom, not price. The problem with the non-free manual is not that publishers charge a price for printed copies—that in itself is fine. (The Free Software Foundation sells printed copies of manuals, too.) The problem is the restrictions on the use of the manual. Free manuals are available in source code form, and give you permission to copy and modify. Non-free manuals do not allow this.

The criteria of freedom for a free manual are roughly the same as for free software. Redistribution (including the normal kinds of commercial redistribution) must be permitted, so that the manual can accompany every copy of the program, both on-line and on paper.

Permission for modification of the technical content is crucial too. When people modify the software, adding or changing features, if they are conscientious they will change the manual too—so they can provide accurate and clear documentation for the modified program. A manual that leaves you no choice but to write a new manual to document a changed version of the program is not really available to our community.

Some kinds of limits on the way modification is handled are acceptable. For example, requirements to preserve the original author’s copyright notice, the distribution terms, or the list of authors, are ok. It is also no problem to require modified versions to include notice that they were modified. Even entire sections that may not be deleted or changed are acceptable, as long as they deal with nontechnical topics (like this one). These kinds of restrictions are acceptable because they don’t obstruct the community’s normal use of the manual.

However, it must be possible to modify all the technical content of the manual, and then distribute the result in all the usual media, through all the usual channels. Otherwise, the restrictions obstruct the use of the manual, it is not free, and we need another manual to replace it.

Please spread the word about this issue. Our community continues to lose manuals to proprietary publishing. If we spread the word that free software needs free reference manuals and free tutorials, perhaps the next person who wants to contribute by writing documentation will realize, before it is too late, that only free manuals contribute to the free software community.

If you are writing documentation, please insist on publishing it under the GNU Free Documentation License or another free documentation license. Remember that this decision requires your approval—you don’t have to let the publisher decide. Some commercial publishers will use a free license if you insist, but they will not propose the option; it is up to you to raise the issue and say firmly that this is what you want. If the publisher you are dealing with refuses, please try other publishers. If you’re not sure whether a proposed license is free, write to licensing@gnu.org.

You can encourage commercial publishers to sell more free, copylefted manuals and tutorials by buying them, and particularly by buying copies from the publishers that paid for their writing or for major improvements. Meanwhile, try to avoid buying non-free documentation at all. Check the distribution terms of a manual before you buy it, and insist that whoever seeks your business must respect your freedom. Check the history of the book, and try reward the publishers that have paid or pay the authors to work on it.

The Free Software Foundation maintains a list of free documentation published by other publishers:

    http://www.fsf.org/doc/other-free-books.html

Appendix D: GNU General Public License 426

GNU General Public License

Version 2, June 1991

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1989, 1991 Free Software Foundation, Inc. 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

Preamble

The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software—to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation’s software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.

To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.

Also, for each author’s protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors’ reputations.

Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone’s free use or not licensed at all.

The precise terms and conditions for copying, distribution and modification follow.

TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION

0. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The “Program”, below, refers to any such program or work, and a “work based on the Program” means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term “modification”.) Each licensee is addressed as “you”.

Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.

1. You may copy and distribute verbatim copies of the Program’s source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program.

You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.

2. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions: a. You must cause the modified files to carry prominent notices stating that you changed the files and the

date of any change. b. You must cause any work that you distribute or publish, that in whole or in part contains or is derived

from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.

c. If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)

These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it. Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program. In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.

3. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable

form under the terms of Sections 1 and 2 above provided that you also do one of the following:

a. Accompany it with the complete corresponding machine-readable source code, which must be distributed

under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or, b. Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no

more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

c. Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)

The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable. If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.

4. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

5. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it.

6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients’ exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.

7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program.

If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances.

It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.

This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.

8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.

9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and “any later version”, you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.

10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.

NO WARRANTY

11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PRO-GRAM “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUD-ING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GEN-ERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAIL-URE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

END OF TERMS AND CONDITIONS Appendix: How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

      one line to give the program's name and a brief idea
      of what it does.
      Copyright (C) yyyy  name of author
      This program is free software; you can redistribute it
      and/or modify it under the terms of the GNU General Public
      License as published by the Free Software Foundation; either
      version 2 of the License, or (at your option) any later
      version.
      This program is distributed in the hope that it will be
      useful, but WITHOUT ANY WARRANTY; without even the implied
      warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
      PURPOSE.  See the GNU General Public License for more
      details.
      You should have received a copy of the GNU General Public
      License along with this program; if not, write to the Free
      Software Foundation, Inc., 59 Temple Place - Suite 330,
      Boston, MA 02111-1307, USA.

Also add information on how to contact you by electronic and paper mail.

If the program is interactive, make it output a short notice like this when it starts in an interactive mode:

      Gnomovision version 69, Copyright (C) 19yy name of author
      Gnomovision comes with ABSOLUTELY NO WARRANTY; for details
      type `show w'.  This is free software, and you are welcome
      to redistribute it under certain conditions; type `show c'
      for details.

The hypothetical commands ‘show w’ and ‘show c’ should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than ‘show w’ and ‘show c’; they could even be mouseclicks or menu items—whatever suits your program.

You should also get your employer (if you work as a programmer) or your school, if any, to sign a “copyright disclaimer” for the program, if necessary. Here is a sample; alter the names:

      Yoyodyne, Inc., hereby disclaims all copyright interest in
      the program `Gnomovision' (which makes passes at compilers)
      written by James Hacker.
      signature of Ty Coon, 1 April 1989
      Ty Coon, President of Vice

This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.

Appendix D: GNU Free Documentation License 430

GNU Free Documentation License

Version 1.2, November 2002

Copyright \begin{matrix}c \\ ◯ \end{matrix} 2000,2001,2002 Free Software Foundation, Inc.

51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

0. PREAMBLE

The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

This License is a kind of “copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

1. APPLICABILITY AND DEFINITIONS

This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royaltyfree license, unlimited in duration, to use that work under the conditions stated herein. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.

A “Modified Version” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

A “Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document’s overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none.

The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.

A “Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not “Transparent” is called “Opaque”.

Examples of suitable formats for Transparent copies include plain ascii without markup, Texinfo input format, LaTEX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.

The “Title Page” means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, “Title Page” means the text near the most prominent appearance of the work’s title, preceding the beginning of the body of the text.

A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”, “Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when you modify the Document means that it remains a section “Entitled XYZ” according to this definition.

The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.

2. VERBATIM COPYING

You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.

You may also lend copies, under the same conditions stated above, and you may publicly display copies.

3. COPYING IN QUANTITY

If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document’s license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using publicstandard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

4. MODIFICATIONS

You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.

B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.

C. State on the Title page the name of the publisher of the Modified Version, as the publisher.

D. Preserve all the copyright notices of the Document.

E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.

F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.

G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document’s license notice.

H. Include an unaltered copy of this License.

I. Preserve the section Entitled “History”, Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled “History” in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.

J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the “History” section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.

K. For any section Entitled “Acknowledgements” or “Dedications”, Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.

L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section

numbers or the equivalent are not considered part of the section titles. M. Delete any section Entitled “Endorsements”. Such a section may not be included in the Modified Version. N. Do not retitle any existing section to be Entitled “Endorsements” or to conflict in title with any Invariant

Section. O. Preserve any Warranty Disclaimers.

If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version’s license notice. These titles must be distinct from any other section titles. You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one. The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

5. COMBINING DOCUMENTS You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections Entitled “History” in the various original documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled “Endorsements.”

6. COLLECTIONS OF DOCUMENTS You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

7. AGGREGATION WITH INDEPENDENT WORKS A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation’s users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

8. TRANSLATION

Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

9. TERMINATION

You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

10. FUTURE REVISIONS OF THIS LICENSE

The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

        Copyright (C) year your name.
        Permission is granted to copy, distribute and/or modify
        this document under the terms of the GNU Free
        Documentation License, Version 1.2 or any later version
        published by the Free Software Foundation; with no
        Invariant Sections, no Front-Cover Texts, and no
        Back-Cover Texts.  A copy of the license is included in
        the section entitled ``GNU Free Documentation License''.

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.” line with this:

        with the Invariant Sections being list their
        titles, with the Front-Cover Texts being list, and
        with the Back-Cover Texts being list.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

Appendix D: Function Index 435

Function Index

C

cblas_caxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_ccopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_cdotc_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_cdotu_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_cgbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_cgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_cgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_cgerc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_cgeru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_chbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_chemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_chemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_cher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_cher2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_cher2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_cherk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_chpmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_chpr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_chpr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_cscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_csscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_cswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_csymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_csyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_csyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_ctbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ctbsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ctpmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ctpsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ctrmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_ctrmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ctrsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_ctrsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_dasum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_daxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_dcopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_ddot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_dgbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_dgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_dnrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_drot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_drotg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_drotm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_drotmg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_dsbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_dscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_dsdot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_dspmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_dspr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_dspr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_dswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_dsymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_dsymv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_dsyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_dsyr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cblas_dsyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_dsyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_dtbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dtbsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dtpmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dtpsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dtrmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_dtrmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dtrsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_dtrsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_dzasum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_dznrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_icamax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_idamax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_isamax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_izamax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_sasum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_saxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_scasum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_scnrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_scopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_sdot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_sdsdot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_sgbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_sgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_sgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_sger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_snrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_srot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_srotg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_srotm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_srotmg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_ssbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_sscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_sspmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_sspr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_sspr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_sswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_ssymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_ssymv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_ssyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_ssyr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_ssyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_ssyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_stbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_stbsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_stpmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_stpsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cblas_strmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_strmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_strsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cblas_strsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_xerbla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 cblas_zaxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_zcopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_zdotc_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_zdotu_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cblas_zdscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_zgbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_zgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 cblas_zgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_zgerc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zgeru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zhbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zhemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_zhemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zher2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zher2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 cblas_zherk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_zhpmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zhpr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zhpr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 cblas_zscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 cblas_zswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cblas_zsymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_zsyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_zsyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_ztbmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ztbsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_ztpmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ztpsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 cblas_ztrmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_ztrmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cblas_ztrsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 cblas_ztrsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

G

gsl_acosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_asinh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_atanh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_blas_caxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_ccopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_cdotc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_cdotu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 gsl_blas_cgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_cgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_cgerc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_cgeru . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_chemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_chemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_cher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_cher2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_cher2k . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_cherk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_cscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_csscal . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_cswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_csymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_csyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_csyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_ctrmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_ctrmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_ctrsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_ctrsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_dasum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_daxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_dcopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_ddot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 gsl_blas_dgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_dgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_dger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_dnrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_drot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_drotg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_drotm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_drotmg . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_dscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_dsdot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 gsl_blas_dswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_dsymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_dsymv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_dsyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_dsyr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_dsyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_dsyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_dtrmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_dtrmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_dtrsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_dtrsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_dzasum . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_dznrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_icamax . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_idamax . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_isamax . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_izamax . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_sasum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_saxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_scasum . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_scnrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_scopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_sdot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 gsl_blas_sdsdot . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 gsl_blas_sgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_sgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_sger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_snrm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_srot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_srotg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_srotm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_srotmg . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_sscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_sswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_ssymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_ssymv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_ssyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_ssyr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_ssyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_ssyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_strmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_strmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_strsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_strsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_zaxpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_zcopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_zdotc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_zdotu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 gsl_blas_zdscal . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_zgemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_zgemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_zgerc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_zgeru . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_zhemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_zhemv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 gsl_blas_zher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_zher2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 gsl_blas_zher2k . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_zherk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_zscal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 gsl_blas_zswap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 gsl_blas_zsymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 gsl_blas_zsyr2k . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 gsl_blas_zsyrk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_ztrmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_ztrmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_blas_ztrsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 gsl_blas_ztrsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 gsl_block_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 gsl_block_calloc . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 gsl_block_fprintf . . . . . . . . . . . . . . . . . . . . . . . . . . 71 gsl_block_fread . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 gsl_block_free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 gsl_block_fscanf . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 gsl_block_fwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 gsl_cdf_beta_P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 gsl_cdf_beta_Pinv . . . . . . . . . . . . . . . . . . . . . . . . . 208 gsl_cdf_beta_Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 gsl_cdf_beta_Qinv . . . . . . . . . . . . . . . . . . . . . . . . . 208 gsl_cdf_binomial_P. . . . . . . . . . . . . . . . . . . . . . . . 220 gsl_cdf_binomial_Q. . . . . . . . . . . . . . . . . . . . . . . . 220 gsl_cdf_cauchy_P . . . . . . . . . . . . . . . . . . . . . . . . . . 196 gsl_cdf_cauchy_Pinv. . . . . . . . . . . . . . . . . . . . . . . 196 gsl_cdf_cauchy_Q . . . . . . . . . . . . . . . . . . . . . . . . . . 196 gsl_cdf_cauchy_Qinv. . . . . . . . . . . . . . . . . . . . . . . 196 gsl_cdf_chisq_P . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 gsl_cdf_chisq_Pinv. . . . . . . . . . . . . . . . . . . . . . . . 205 gsl_cdf_chisq_Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 gsl_cdf_chisq_Qinv. . . . . . . . . . . . . . . . . . . . . . . . 205 gsl_cdf_exponential_P . . . . . . . . . . . . . . . . . . . . 193

gsl_cdf_exponential_Pinv . . . . . . . . . . . . . . . . . 193

gsl_cdf_exponential_Q . . . . . . . . . . . . . . . . . . . . 193

gsl_cdf_exponential_Qinv . . . . . . . . . . . . . . . . . 193

gsl_cdf_exppow_P . . . . . . . . . . . . . . . . . . . . . . . . . . 195 gsl_cdf_exppow_Q . . . . . . . . . . . . . . . . . . . . . . . . . . 195 gsl_cdf_fdist_P . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 gsl_cdf_fdist_Pinv. . . . . . . . . . . . . . . . . . . . . . . . 206 gsl_cdf_fdist_Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 gsl_cdf_fdist_Qinv. . . . . . . . . . . . . . . . . . . . . . . . 206 gsl_cdf_flat_P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 gsl_cdf_flat_Pinv . . . . . . . . . . . . . . . . . . . . . . . . . 203 gsl_cdf_flat_Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 gsl_cdf_flat_Qinv . . . . . . . . . . . . . . . . . . . . . . . . . 203 gsl_cdf_gamma_P . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 gsl_cdf_gamma_Pinv. . . . . . . . . . . . . . . . . . . . . . . . 202 gsl_cdf_gamma_Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 gsl_cdf_gamma_Qinv. . . . . . . . . . . . . . . . . . . . . . . . 202 gsl_cdf_gaussian_P. . . . . . . . . . . . . . . . . . . . . . . . 190 gsl_cdf_gaussian_Pinv . . . . . . . . . . . . . . . . . . . . 190 gsl_cdf_gaussian_Q. . . . . . . . . . . . . . . . . . . . . . . . 190 gsl_cdf_gaussian_Qinv . . . . . . . . . . . . . . . . . . . . 190 gsl_cdf_geometric_P. . . . . . . . . . . . . . . . . . . . . . . 224 gsl_cdf_geometric_Q. . . . . . . . . . . . . . . . . . . . . . . 224 gsl_cdf_gumbel1_P . . . . . . . . . . . . . . . . . . . . . . . . . 213 gsl_cdf_gumbel1_Pinv. . . . . . . . . . . . . . . . . . . . . . 213 gsl_cdf_gumbel1_Q . . . . . . . . . . . . . . . . . . . . . . . . . 213 gsl_cdf_gumbel1_Qinv. . . . . . . . . . . . . . . . . . . . . . 213 gsl_cdf_gumbel2_P . . . . . . . . . . . . . . . . . . . . . . . . . 214 gsl_cdf_gumbel2_Pinv. . . . . . . . . . . . . . . . . . . . . . 214 gsl_cdf_gumbel2_Q . . . . . . . . . . . . . . . . . . . . . . . . . 214 gsl_cdf_gumbel2_Qinv. . . . . . . . . . . . . . . . . . . . . . 214

                                                gsl_cdf_hypergeometric_P . . . . . . . . . . . . . . . . . 225
                                                gsl_cdf_hypergeometric_Q . . . . . . . . . . . . . . . . . 225

gsl_cdf_laplace_P . . . . . . . . . . . . . . . . . . . . . . . . . 194 gsl_cdf_laplace_Pinv. . . . . . . . . . . . . . . . . . . . . . 194 gsl_cdf_laplace_Q . . . . . . . . . . . . . . . . . . . . . . . . . 194 gsl_cdf_laplace_Qinv. . . . . . . . . . . . . . . . . . . . . . 194 gsl_cdf_logistic_P. . . . . . . . . . . . . . . . . . . . . . . . 209 gsl_cdf_logistic_Pinv . . . . . . . . . . . . . . . . . . . . 209 gsl_cdf_logistic_Q. . . . . . . . . . . . . . . . . . . . . . . . 209 gsl_cdf_logistic_Qinv . . . . . . . . . . . . . . . . . . . . 209 gsl_cdf_lognormal_P. . . . . . . . . . . . . . . . . . . . . . . 204

                                                gsl_cdf_lognormal_Pinv . . . . . . . . . . . . . . . . . . . 204

gsl_cdf_lognormal_Q. . . . . . . . . . . . . . . . . . . . . . . 204

                                                gsl_cdf_lognormal_Qinv . . . . . . . . . . . . . . . . . . . 204
                                                gsl_cdf_negative_binomial_P . . . . . . . . . . . . . . 222
                                                gsl_cdf_negative_binomial_Q . . . . . . . . . . . . . . 222

gsl_cdf_pareto_P . . . . . . . . . . . . . . . . . . . . . . . . . . 210 gsl_cdf_pareto_Pinv. . . . . . . . . . . . . . . . . . . . . . . 210 gsl_cdf_pareto_Q . . . . . . . . . . . . . . . . . . . . . . . . . . 210 gsl_cdf_pareto_Qinv. . . . . . . . . . . . . . . . . . . . . . . 210 gsl_cdf_pascal_P . . . . . . . . . . . . . . . . . . . . . . . . . . 223 gsl_cdf_pascal_Q . . . . . . . . . . . . . . . . . . . . . . . . . . 223 gsl_cdf_poisson_P . . . . . . . . . . . . . . . . . . . . . . . . . 218 gsl_cdf_poisson_Q . . . . . . . . . . . . . . . . . . . . . . . . . 218 gsl_cdf_rayleigh_P. . . . . . . . . . . . . . . . . . . . . . . . 197 gsl_cdf_rayleigh_Pinv . . . . . . . . . . . . . . . . . . . . 197 gsl_cdf_rayleigh_Q. . . . . . . . . . . . . . . . . . . . . . . . 197 gsl_cdf_rayleigh_Qinv . . . . . . . . . . . . . . . . . . . . 197 gsl_cdf_tdist_P . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 gsl_cdf_tdist_Pinv. . . . . . . . . . . . . . . . . . . . . . . . 207 gsl_cdf_tdist_Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 gsl_cdf_tdist_Qinv. . . . . . . . . . . . . . . . . . . . . . . . 207 gsl_cdf_ugaussian_P. . . . . . . . . . . . . . . . . . . . . . . 190

                                                gsl_cdf_ugaussian_Pinv . . . . . . . . . . . . . . . . . . . 190

gsl_cdf_ugaussian_Q. . . . . . . . . . . . . . . . . . . . . . . 190

                                                gsl_cdf_ugaussian_Qinv . . . . . . . . . . . . . . . . . . . 190

gsl_cdf_weibull_P . . . . . . . . . . . . . . . . . . . . . . . . . 212 gsl_cdf_weibull_Pinv. . . . . . . . . . . . . . . . . . . . . . 212 gsl_cdf_weibull_Q . . . . . . . . . . . . . . . . . . . . . . . . . 212 gsl_cdf_weibull_Qinv. . . . . . . . . . . . . . . . . . . . . . 212 gsl_cheb_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 gsl_cheb_calc_deriv. . . . . . . . . . . . . . . . . . . . . . . 305 gsl_cheb_calc_integ. . . . . . . . . . . . . . . . . . . . . . . 305 gsl_cheb_eval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 gsl_cheb_eval_err . . . . . . . . . . . . . . . . . . . . . . . . . 305 gsl_cheb_eval_n . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 gsl_cheb_eval_n_err. . . . . . . . . . . . . . . . . . . . . . . 305 gsl_cheb_free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 gsl_cheb_init . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 gsl_combination_alloc . . . . . . . . . . . . . . . . . . . . 100

gsl_combination_calloc . . . . . . . . . . . . . . . . . . . 100

gsl_combination_data. . . . . . . . . . . . . . . . . . . . . . 101

gsl_combination_fprintf . . . . . . . . . . . . . . . . . . 102

gsl_combination_fread . . . . . . . . . . . . . . . . . . . . 102 gsl_combination_free. . . . . . . . . . . . . . . . . . . . . . 100

gsl_combination_fscanf . . . . . . . . . . . . . . . . . . . 102
gsl_combination_fwrite . . . . . . . . . . . . . . . . . . . 102

gsl_combination_get. . . . . . . . . . . . . . . . . . . . . . . 101

gsl_combination_init_first . . . . . . . . . . . . . . . 100
gsl_combination_init_last . . . . . . . . . . . . . . . . 100

gsl_combination_k . . . . . . . . . . . . . . . . . . . . . . . . . 101

gsl_combination_memcpy . . . . . . . . . . . . . . . . . . . 101

gsl_combination_n . . . . . . . . . . . . . . . . . . . . . . . . . 101 gsl_combination_next. . . . . . . . . . . . . . . . . . . . . . 101 gsl_combination_prev. . . . . . . . . . . . . . . . . . . . . . 101 gsl_combination_valid . . . . . . . . . . . . . . . . . . . . 101 gsl_complex_abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_abs2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_add_imag. . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_add_real. . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_arccos . . . . . . . . . . . . . . . . . . . . . . . . . 24

gsl_complex_arccos_real . . . . . . . . . . . . . . . . . . . 24

gsl_complex_arccosh. . . . . . . . . . . . . . . . . . . . . . . . 25

gsl_complex_arccosh_real . . . . . . . . . . . . . . . . . . 25

gsl_complex_arccot . . . . . . . . . . . . . . . . . . . . . . . . . 24 gsl_complex_arccoth. . . . . . . . . . . . . . . . . . . . . . . . 25 gsl_complex_arccsc . . . . . . . . . . . . . . . . . . . . . . . . . 24

gsl_complex_arccsc_real . . . . . . . . . . . . . . . . . . . 24

gsl_complex_arccsch. . . . . . . . . . . . . . . . . . . . . . . . 25 gsl_complex_arcsec . . . . . . . . . . . . . . . . . . . . . . . . . 24

gsl_complex_arcsec_real . . . . . . . . . . . . . . . . . . . 24

gsl_complex_arcsech. . . . . . . . . . . . . . . . . . . . . . . . 25 gsl_complex_arcsin . . . . . . . . . . . . . . . . . . . . . . . . . 23

gsl_complex_arcsin_real . . . . . . . . . . . . . . . . . . . 23

gsl_complex_arcsinh. . . . . . . . . . . . . . . . . . . . . . . . 25 gsl_complex_arctan . . . . . . . . . . . . . . . . . . . . . . . . . 24 gsl_complex_arctanh. . . . . . . . . . . . . . . . . . . . . . . . 25

gsl_complex_arctanh_real . . . . . . . . . . . . . . . . . . 25

gsl_complex_arg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_conjugate . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_cos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_cosh . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 gsl_complex_cot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_coth . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 gsl_complex_csc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_csch . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 gsl_complex_div . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_div_imag. . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_div_real. . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_inverse. . . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_log . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_log_b . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_log10 . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_logabs . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_mul_imag. . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_mul_real. . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_negative. . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_polar . . . . . . . . . . . . . . . . . . . . . . . . . . 20 gsl_complex_pow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_pow_real. . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_rect . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 gsl_complex_sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_sech . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 gsl_complex_sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_sinh . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 gsl_complex_sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_sqrt_real . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_sub_imag. . . . . . . . . . . . . . . . . . . . . . . 22 gsl_complex_sub_real. . . . . . . . . . . . . . . . . . . . . . . 21 gsl_complex_tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 gsl_complex_tanh . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 gsl_deriv_backward. . . . . . . . . . . . . . . . . . . . . . . . 301 gsl_deriv_central . . . . . . . . . . . . . . . . . . . . . . . . . 301 gsl_deriv_forward . . . . . . . . . . . . . . . . . . . . . . . . . 301 gsl_dht_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 gsl_dht_apply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 gsl_dht_free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 gsl_dht_init . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 gsl_dht_k_sample . . . . . . . . . . . . . . . . . . . . . . . . . . 320 gsl_dht_new . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 gsl_dht_x_sample . . . . . . . . . . . . . . . . . . . . . . . . . . 320 GSL_EDOM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 gsl_eigen_herm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 gsl_eigen_herm_alloc. . . . . . . . . . . . . . . . . . . . . . 136 gsl_eigen_herm_free. . . . . . . . . . . . . . . . . . . . . . . 136 gsl_eigen_hermv . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 gsl_eigen_hermv_alloc . . . . . . . . . . . . . . . . . . . . 136 gsl_eigen_hermv_free. . . . . . . . . . . . . . . . . . . . . . 136 gsl_eigen_hermv_sort. . . . . . . . . . . . . . . . . . . . . . 137 gsl_eigen_symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 gsl_eigen_symm_alloc. . . . . . . . . . . . . . . . . . . . . . 135 gsl_eigen_symm_free. . . . . . . . . . . . . . . . . . . . . . . 135 gsl_eigen_symmv . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 gsl_eigen_symmv_alloc . . . . . . . . . . . . . . . . . . . . 135 gsl_eigen_symmv_free. . . . . . . . . . . . . . . . . . . . . . 135 gsl_eigen_symmv_sort. . . . . . . . . . . . . . . . . . . . . . 136 GSL_EINVAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 GSL_ENOMEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 GSL_ERANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 GSL_ERROR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 GSL_ERROR_VAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 gsl_expm1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_fcmp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

                                                gsl_fft_complex_backward . . . . . . . . . . . . . . . . . 146
                                                gsl_fft_complex_forward . . . . . . . . . . . . . . . . . . 146
                                                gsl_fft_complex_inverse . . . . . . . . . . . . . . . . . . 146
                                                gsl_fft_complex_radix2_backward. . . . . . . . . . 142
                                                gsl_fft_complex_radix2_dif_backward . . . . . 142
                                                gsl_fft_complex_radix2_dif_forward . . . . . . 142
                                                gsl_fft_complex_radix2_dif_inverse . . . . . . 142
                                                gsl_fft_complex_radix2_dif_transform . . . . 142
                                                gsl_fft_complex_radix2_forward . . . . . . . . . . . 142
                                                gsl_fft_complex_radix2_inverse . . . . . . . . . . . 142
                                                gsl_fft_complex_radix2_transform. . . . . . . . . 142
                                                gsl_fft_complex_transform . . . . . . . . . . . . . . . . 146
                                                gsl_fft_complex_wavetable_alloc. . . . . . . . . . 145
                                                gsl_fft_complex_wavetable_free . . . . . . . . . . . 145

gsl_fft_complex_workspace_alloc. . . . . . . . . . 146
gsl_fft_complex_workspace_free . . . . . . . . . . . 146
gsl_fft_halfcomplex_radix2_backward . . . . . 150
gsl_fft_halfcomplex_radix2_inverse . . . . . . 150
gsl_fft_halfcomplex_transform . . . . . . . . . . . . 152
gsl_fft_halfcomplex_unpack . . . . . . . . . . . . . . . 153
gsl_fft_halfcomplex_wavetable_alloc . . . . . 151
gsl_fft_halfcomplex_wavetable_free . . . . . . 151
gsl_fft_real_radix2_transform . . . . . . . . . . . . 149
gsl_fft_real_transform . . . . . . . . . . . . . . . . . . . 152

gsl_fft_real_unpack. . . . . . . . . . . . . . . . . . . . . . . 152

gsl_fft_real_wavetable_alloc . . . . . . . . . . . . . 151
gsl_fft_real_wavetable_free . . . . . . . . . . . . . . 151
gsl_fft_real_workspace_alloc . . . . . . . . . . . . . 152
gsl_fft_real_workspace_free . . . . . . . . . . . . . . 152

gsl_finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_fit_linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 gsl_fit_linear_est. . . . . . . . . . . . . . . . . . . . . . . . 367 gsl_fit_mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 gsl_fit_mul_est . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 gsl_fit_wlinear . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 gsl_fit_wmul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 gsl_frexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_heapsort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 gsl_heapsort_index. . . . . . . . . . . . . . . . . . . . . . . . 105

gsl_histogram_accumulate . . . . . . . . . . . . . . . . . 244

gsl_histogram_add . . . . . . . . . . . . . . . . . . . . . . . . . 246 gsl_histogram_alloc. . . . . . . . . . . . . . . . . . . . . . . 243 gsl_histogram_bins. . . . . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_clone. . . . . . . . . . . . . . . . . . . . . . . 244 gsl_histogram_div . . . . . . . . . . . . . . . . . . . . . . . . . 246

gsl_histogram_equal_bins_p . . . . . . . . . . . . . . . 246

gsl_histogram_find. . . . . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_fprintf . . . . . . . . . . . . . . . . . . . . 247 gsl_histogram_fread. . . . . . . . . . . . . . . . . . . . . . . 247 gsl_histogram_free. . . . . . . . . . . . . . . . . . . . . . . . 244 gsl_histogram_fscanf. . . . . . . . . . . . . . . . . . . . . . 247 gsl_histogram_fwrite. . . . . . . . . . . . . . . . . . . . . . 246 gsl_histogram_get . . . . . . . . . . . . . . . . . . . . . . . . . 244

gsl_histogram_get_range . . . . . . . . . . . . . . . . . . 244
gsl_histogram_increment . . . . . . . . . . . . . . . . . . 244

gsl_histogram_max . . . . . . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_max_bin . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_max_val . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_mean. . . . . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_memcpy. . . . . . . . . . . . . . . . . . . . . . 244 gsl_histogram_min . . . . . . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_min_bin . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_min_val . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_mul . . . . . . . . . . . . . . . . . . . . . . . . . 246

gsl_histogram_pdf_alloc . . . . . . . . . . . . . . . . . . 248
gsl_histogram_pdf_free . . . . . . . . . . . . . . . . . . . 248
gsl_histogram_pdf_init . . . . . . . . . . . . . . . . . . . 248
gsl_histogram_pdf_sample . . . . . . . . . . . . . . . . . 248

gsl_histogram_reset. . . . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_scale. . . . . . . . . . . . . . . . . . . . . . . 246

gsl_histogram_set_ranges . . . . . . . . . . . . . . . . . 243
gsl_histogram_set_ranges_uniform. . . . . . . . . 243

gsl_histogram_shift. . . . . . . . . . . . . . . . . . . . . . . 246 gsl_histogram_sigma. . . . . . . . . . . . . . . . . . . . . . . 245 gsl_histogram_sub . . . . . . . . . . . . . . . . . . . . . . . . . 246 gsl_histogram_sum . . . . . . . . . . . . . . . . . . . . . . . . . 246

                                                gsl_histogram2d_accumulate . . . . . . . . . . . . . . . 252

gsl_histogram2d_add. . . . . . . . . . . . . . . . . . . . . . . 254 gsl_histogram2d_alloc . . . . . . . . . . . . . . . . . . . . 251 gsl_histogram2d_clone . . . . . . . . . . . . . . . . . . . . 252 gsl_histogram2d_cov. . . . . . . . . . . . . . . . . . . . . . . 254 gsl_histogram2d_div. . . . . . . . . . . . . . . . . . . . . . . 255

                                                gsl_histogram2d_equal_bins_p . . . . . . . . . . . . . 254

gsl_histogram2d_find. . . . . . . . . . . . . . . . . . . . . . 253

                                                gsl_histogram2d_fprintf . . . . . . . . . . . . . . . . . . 255

gsl_histogram2d_fread . . . . . . . . . . . . . . . . . . . . 255 gsl_histogram2d_free. . . . . . . . . . . . . . . . . . . . . . 252

                                                gsl_histogram2d_fscanf . . . . . . . . . . . . . . . . . . . 256
                                                gsl_histogram2d_fwrite . . . . . . . . . . . . . . . . . . . 255

gsl_histogram2d_get. . . . . . . . . . . . . . . . . . . . . . . 252

                                                gsl_histogram2d_get_xrange . . . . . . . . . . . . . . . 252
                                                gsl_histogram2d_get_yrange . . . . . . . . . . . . . . . 252
                                                gsl_histogram2d_increment . . . . . . . . . . . . . . . . 252
                                                gsl_histogram2d_max_bin . . . . . . . . . . . . . . . . . . 253
                                                gsl_histogram2d_max_val . . . . . . . . . . . . . . . . . . 253
                                                gsl_histogram2d_memcpy . . . . . . . . . . . . . . . . . . . 252
                                                gsl_histogram2d_min_bin . . . . . . . . . . . . . . . . . . 254
                                                gsl_histogram2d_min_val . . . . . . . . . . . . . . . . . . 253

gsl_histogram2d_mul. . . . . . . . . . . . . . . . . . . . . . . 255 gsl_histogram2d_nx. . . . . . . . . . . . . . . . . . . . . . . . 253 gsl_histogram2d_ny. . . . . . . . . . . . . . . . . . . . . . . . 253

                                                gsl_histogram2d_pdf_alloc . . . . . . . . . . . . . . . . 257
                                                gsl_histogram2d_pdf_free . . . . . . . . . . . . . . . . . 257
                                                gsl_histogram2d_pdf_init . . . . . . . . . . . . . . . . . 257
                                                gsl_histogram2d_pdf_sample . . . . . . . . . . . . . . . 257

gsl_histogram2d_reset . . . . . . . . . . . . . . . . . . . . 253 gsl_histogram2d_scale . . . . . . . . . . . . . . . . . . . . 255

                                                gsl_histogram2d_set_ranges . . . . . . . . . . . . . . . 251
                                                gsl_histogram2d_set_ranges_uniform . . . . . . 251

gsl_histogram2d_shift . . . . . . . . . . . . . . . . . . . . 255 gsl_histogram2d_sub. . . . . . . . . . . . . . . . . . . . . . . 255 gsl_histogram2d_sum. . . . . . . . . . . . . . . . . . . . . . . 254 gsl_histogram2d_xmax. . . . . . . . . . . . . . . . . . . . . . 253 gsl_histogram2d_xmean . . . . . . . . . . . . . . . . . . . . 254 gsl_histogram2d_xmin. . . . . . . . . . . . . . . . . . . . . . 253

                                                gsl_histogram2d_xsigma . . . . . . . . . . . . . . . . . . . 254

gsl_histogram2d_ymax. . . . . . . . . . . . . . . . . . . . . . 253 gsl_histogram2d_ymean . . . . . . . . . . . . . . . . . . . . 254 gsl_histogram2d_ymin. . . . . . . . . . . . . . . . . . . . . . 253

                                                gsl_histogram2d_ysigma . . . . . . . . . . . . . . . . . . . 254

gsl_hypot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_ieee_env_setup. . . . . . . . . . . . . . . . . . . . . . . . 399

                                                gsl_ieee_fprintf_double . . . . . . . . . . . . . . . . . . 397
                                                gsl_ieee_fprintf_float . . . . . . . . . . . . . . . . . . . 397
                                                gsl_ieee_printf_double . . . . . . . . . . . . . . . . . . . 398

gsl_ieee_printf_float . . . . . . . . . . . . . . . . . . . . 398 GSL_IMAG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 gsl_integration_qag. . . . . . . . . . . . . . . . . . . . . . . 159 gsl_integration_qagi. . . . . . . . . . . . . . . . . . . . . . 160 gsl_integration_qagil . . . . . . . . . . . . . . . . . . . . 161 gsl_integration_qagiu . . . . . . . . . . . . . . . . . . . . 161 gsl_integration_qagp. . . . . . . . . . . . . . . . . . . . . . 160 gsl_integration_qags. . . . . . . . . . . . . . . . . . . . . . 160 gsl_integration_qawc. . . . . . . . . . . . . . . . . . . . . . 161 gsl_integration_qawf. . . . . . . . . . . . . . . . . . . . . . 164 gsl_integration_qawo. . . . . . . . . . . . . . . . . . . . . . 163

gsl_integration_qawo_table_alloc. . . . . . . . . 163
gsl_integration_qawo_table_free. . . . . . . . . . 163
gsl_integration_qawo_table_set . . . . . . . . . . . 163
gsl_integration_qawo_table_set_length . . . 163

gsl_integration_qaws. . . . . . . . . . . . . . . . . . . . . . 162

gsl_integration_qaws_table_alloc. . . . . . . . . 162
gsl_integration_qaws_table_free. . . . . . . . . . 162
gsl_integration_qaws_table_set . . . . . . . . . . . 162

gsl_integration_qng. . . . . . . . . . . . . . . . . . . . . . . 158

gsl_integration_workspace_alloc. . . . . . . . . . 159
gsl_integration_workspace_free . . . . . . . . . . . 159
gsl_interp_accel_alloc . . . . . . . . . . . . . . . . . . . 295

gsl_interp_accel_find . . . . . . . . . . . . . . . . . . . . 296 gsl_interp_accel_free . . . . . . . . . . . . . . . . . . . . 296 gsl_interp_akima . . . . . . . . . . . . . . . . . . . . . . . . . . 295

gsl_interp_akima_periodic . . . . . . . . . . . . . . . . 295

gsl_interp_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . 294 gsl_interp_bsearch. . . . . . . . . . . . . . . . . . . . . . . . 295 gsl_interp_cspline. . . . . . . . . . . . . . . . . . . . . . . . 295

gsl_interp_cspline_periodic . . . . . . . . . . . . . . 295

gsl_interp_eval . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 gsl_interp_eval_deriv . . . . . . . . . . . . . . . . . . . . 296

gsl_interp_eval_deriv_e . . . . . . . . . . . . . . . . . . 296
gsl_interp_eval_deriv2 . . . . . . . . . . . . . . . . . . . 296
gsl_interp_eval_deriv2_e . . . . . . . . . . . . . . . . . 296

gsl_interp_eval_e . . . . . . . . . . . . . . . . . . . . . . . . . 296 gsl_interp_eval_integ . . . . . . . . . . . . . . . . . . . . 296

gsl_interp_eval_integ_e . . . . . . . . . . . . . . . . . . 296

gsl_interp_free . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 gsl_interp_init . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 gsl_interp_linear . . . . . . . . . . . . . . . . . . . . . . . . . 294 gsl_interp_min_size. . . . . . . . . . . . . . . . . . . . . . . 295 gsl_interp_name . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 gsl_interp_polynomial . . . . . . . . . . . . . . . . . . . . 294 GSL_IS_EVEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 GSL_IS_ODD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_isinf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_isnan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_ldexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

gsl_linalg_bidiag_decomp . . . . . . . . . . . . . . . . . 130
gsl_linalg_bidiag_unpack . . . . . . . . . . . . . . . . . 130
gsl_linalg_bidiag_unpack_B . . . . . . . . . . . . . . . 130
gsl_linalg_bidiag_unpack2 . . . . . . . . . . . . . . . . 130
gsl_linalg_cholesky_decomp . . . . . . . . . . . . . . . 128
gsl_linalg_cholesky_solve . . . . . . . . . . . . . . . . 128
gsl_linalg_cholesky_svx . . . . . . . . . . . . . . . . . . 128
gsl_linalg_complex_LU_decomp . . . . . . . . . . . . . 122
gsl_linalg_complex_LU_det . . . . . . . . . . . . . . . . 123
gsl_linalg_complex_LU_invert . . . . . . . . . . . . . 123
gsl_linalg_complex_LU_lndet . . . . . . . . . . . . . . 123
gsl_linalg_complex_LU_refine . . . . . . . . . . . . . 123
gsl_linalg_complex_LU_sgndet . . . . . . . . . . . . . 123
gsl_linalg_complex_LU_solve . . . . . . . . . . . . . . 122
gsl_linalg_complex_LU_svx . . . . . . . . . . . . . . . . 122
gsl_linalg_hermtd_decomp . . . . . . . . . . . . . . . . . 129
gsl_linalg_hermtd_unpack . . . . . . . . . . . . . . . . . 129
gsl_linalg_hermtd_unpack_T . . . . . . . . . . . . . . . 129

gsl_linalg_HH_solve. . . . . . . . . . . . . . . . . . . . . . . 131 gsl_linalg_HH_svx . . . . . . . . . . . . . . . . . . . . . . . . . 131

gsl_linalg_householder_hm . . . . . . . . . . . . . . . . 131
gsl_linalg_householder_hv . . . . . . . . . . . . . . . . 131

                                                gsl_linalg_householder_mh . . . . . . . . . . . . . . . . 131
                                                gsl_linalg_householder_transform. . . . . . . . . 131

gsl_linalg_LU_decomp. . . . . . . . . . . . . . . . . . . . . . 122 gsl_linalg_LU_det . . . . . . . . . . . . . . . . . . . . . . . . . 123 gsl_linalg_LU_invert. . . . . . . . . . . . . . . . . . . . . . 123 gsl_linalg_LU_lndet. . . . . . . . . . . . . . . . . . . . . . . 123 gsl_linalg_LU_refine. . . . . . . . . . . . . . . . . . . . . . 123 gsl_linalg_LU_sgndet. . . . . . . . . . . . . . . . . . . . . . 123 gsl_linalg_LU_solve. . . . . . . . . . . . . . . . . . . . . . . 122 gsl_linalg_LU_svx . . . . . . . . . . . . . . . . . . . . . . . . . 122 gsl_linalg_QR_decomp. . . . . . . . . . . . . . . . . . . . . . 124 gsl_linalg_QR_lssolve . . . . . . . . . . . . . . . . . . . . 124 gsl_linalg_QR_QRsolve . . . . . . . . . . . . . . . . . . . . 125 gsl_linalg_QR_QTvec. . . . . . . . . . . . . . . . . . . . . . . 124 gsl_linalg_QR_Qvec. . . . . . . . . . . . . . . . . . . . . . . . 125 gsl_linalg_QR_Rsolve. . . . . . . . . . . . . . . . . . . . . . 125 gsl_linalg_QR_Rsvx. . . . . . . . . . . . . . . . . . . . . . . . 125 gsl_linalg_QR_solve. . . . . . . . . . . . . . . . . . . . . . . 124 gsl_linalg_QR_svx . . . . . . . . . . . . . . . . . . . . . . . . . 124 gsl_linalg_QR_unpack. . . . . . . . . . . . . . . . . . . . . . 125 gsl_linalg_QR_update. . . . . . . . . . . . . . . . . . . . . . 125

                                                gsl_linalg_QRPT_decomp . . . . . . . . . . . . . . . . . . . 126
                                                gsl_linalg_QRPT_decomp2 . . . . . . . . . . . . . . . . . . 126
                                                gsl_linalg_QRPT_QRsolve . . . . . . . . . . . . . . . . . . 126
                                                gsl_linalg_QRPT_Rsolve . . . . . . . . . . . . . . . . . . . 127

gsl_linalg_QRPT_Rsvx. . . . . . . . . . . . . . . . . . . . . . 127 gsl_linalg_QRPT_solve . . . . . . . . . . . . . . . . . . . . 126 gsl_linalg_QRPT_svx. . . . . . . . . . . . . . . . . . . . . . . 126

                                                gsl_linalg_QRPT_update . . . . . . . . . . . . . . . . . . . 126

gsl_linalg_R_solve. . . . . . . . . . . . . . . . . . . . . . . . 125 gsl_linalg_R_svx . . . . . . . . . . . . . . . . . . . . . . . . . . 125

                                                gsl_linalg_solve_cyc_tridiag . . . . . . . . . . . . . 132
                                                gsl_linalg_solve_symm_cyc_tridiag . . . . . . . 132
                                                gsl_linalg_solve_symm_tridiag . . . . . . . . . . . . 132
                                                gsl_linalg_solve_tridiag . . . . . . . . . . . . . . . . . 131

gsl_linalg_SV_decomp. . . . . . . . . . . . . . . . . . . . . . 127

                                                gsl_linalg_SV_decomp_jacobi . . . . . . . . . . . . . . 127
                                                gsl_linalg_SV_decomp_mod . . . . . . . . . . . . . . . . . 127

gsl_linalg_SV_solve. . . . . . . . . . . . . . . . . . . . . . . 128

                                                gsl_linalg_symmtd_decomp . . . . . . . . . . . . . . . . . 129
                                                gsl_linalg_symmtd_unpack . . . . . . . . . . . . . . . . . 129
                                                gsl_linalg_symmtd_unpack_T . . . . . . . . . . . . . . . 129

gsl_log1p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gsl_matrix_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

                                                gsl_matrix_add_constant . . . . . . . . . . . . . . . . . . . 88

gsl_matrix_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 gsl_matrix_calloc . . . . . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_column . . . . . . . . . . . . . . . . . . . . . . . . . . 86

                                                gsl_matrix_const_column . . . . . . . . . . . . . . . . . . . 86
                                                gsl_matrix_const_diagonal . . . . . . . . . . . . . . . . . 86

gsl_matrix_const_ptr. . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_const_row. . . . . . . . . . . . . . . . . . . . . . . 85

                                                gsl_matrix_const_subdiagonal . . . . . . . . . . . . . . 86
                                                gsl_matrix_const_submatrix . . . . . . . . . . . . . . . . 83
                                                gsl_matrix_const_superdiagonal . . . . . . . . . . . . 86
                                                gsl_matrix_const_view_array . . . . . . . . . . . . . . . 84
                                                gsl_matrix_const_view_array_with_tda . . . . . 84
                                                gsl_matrix_const_view_vector . . . . . . . . . . . . . . 85
                                                gsl_matrix_const_view_vector_with_tda . . . . 85

gsl_matrix_diagonal. . . . . . . . . . . . . . . . . . . . . . . . 86

                                                gsl_matrix_div_elements . . . . . . . . . . . . . . . . . . . 88

gsl_matrix_fprintf . . . . . . . . . . . . . . . . . . . . . . . . . 83 gsl_matrix_fread . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 gsl_matrix_free . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_fscanf . . . . . . . . . . . . . . . . . . . . . . . . . . 83 gsl_matrix_fwrite . . . . . . . . . . . . . . . . . . . . . . . . . . 83 gsl_matrix_get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_get_col . . . . . . . . . . . . . . . . . . . . . . . . . 87 gsl_matrix_get_row . . . . . . . . . . . . . . . . . . . . . . . . . 87 gsl_matrix_isnull . . . . . . . . . . . . . . . . . . . . . . . . . . 89 gsl_matrix_max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 gsl_matrix_max_index. . . . . . . . . . . . . . . . . . . . . . . 88 gsl_matrix_memcpy . . . . . . . . . . . . . . . . . . . . . . . . . . 86 gsl_matrix_min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 gsl_matrix_min_index. . . . . . . . . . . . . . . . . . . . . . . 88 gsl_matrix_minmax . . . . . . . . . . . . . . . . . . . . . . . . . . 88

gsl_matrix_minmax_index . . . . . . . . . . . . . . . . . . . 89
gsl_matrix_mul_elements . . . . . . . . . . . . . . . . . . . 88

gsl_matrix_ptr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 gsl_matrix_scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 gsl_matrix_set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_set_all . . . . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_set_col . . . . . . . . . . . . . . . . . . . . . . . . . 87

gsl_matrix_set_identity . . . . . . . . . . . . . . . . . . . 82

gsl_matrix_set_row . . . . . . . . . . . . . . . . . . . . . . . . . 87 gsl_matrix_set_zero. . . . . . . . . . . . . . . . . . . . . . . . 82 gsl_matrix_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

gsl_matrix_subdiagonal . . . . . . . . . . . . . . . . . . . . 86

gsl_matrix_submatrix. . . . . . . . . . . . . . . . . . . . . . . 83

gsl_matrix_superdiagonal . . . . . . . . . . . . . . . . . . 86

gsl_matrix_swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

gsl_matrix_swap_columns . . . . . . . . . . . . . . . . . . . 87
gsl_matrix_swap_rowcol . . . . . . . . . . . . . . . . . . . . 87

gsl_matrix_swap_rows. . . . . . . . . . . . . . . . . . . . . . . 87 gsl_matrix_transpose. . . . . . . . . . . . . . . . . . . . . . . 87

gsl_matrix_transpose_memcpy . . . . . . . . . . . . . . . 87

gsl_matrix_view_array . . . . . . . . . . . . . . . . . . . . . 84

gsl_matrix_view_array_with_tda . . . . . . . . . . . . 84
gsl_matrix_view_vector . . . . . . . . . . . . . . . . . . . . 85
gsl_matrix_view_vector_with_tda . . . . . . . . . . . 85

GSL_MAX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 GSL_MAX_DBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 GSL_MAX_INT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 GSL_MAX_LDBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 GSL_MIN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 GSL_MIN_DBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

gsl_min_fminimizer_alloc . . . . . . . . . . . . . . . . . 335
gsl_min_fminimizer_brent . . . . . . . . . . . . . . . . . 338
gsl_min_fminimizer_f_lower . . . . . . . . . . . . . . . 337
gsl_min_fminimizer_f_minimum . . . . . . . . . . . . . 337
gsl_min_fminimizer_f_upper . . . . . . . . . . . . . . . 337
gsl_min_fminimizer_free . . . . . . . . . . . . . . . . . . 336
gsl_min_fminimizer_goldensection. . . . . . . . . 338
gsl_min_fminimizer_iterate . . . . . . . . . . . . . . . 336
gsl_min_fminimizer_name . . . . . . . . . . . . . . . . . . 336
gsl_min_fminimizer_set . . . . . . . . . . . . . . . . . . . 335
gsl_min_fminimizer_set_with_values . . . . . . 336
gsl_min_fminimizer_x_lower . . . . . . . . . . . . . . . 337
gsl_min_fminimizer_x_minimum . . . . . . . . . . . . . 336
gsl_min_fminimizer_x_upper . . . . . . . . . . . . . . . 337

GSL_MIN_INT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 GSL_MIN_LDBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 gsl_min_test_interval . . . . . . . . . . . . . . . . . . . . 337 gsl_monte_miser_alloc . . . . . . . . . . . . . . . . . . . . 269 gsl_monte_miser_free. . . . . . . . . . . . . . . . . . . . . . 269 gsl_monte_miser_init. . . . . . . . . . . . . . . . . . . . . . 269

                                                gsl_monte_miser_integrate . . . . . . . . . . . . . . . . 269

gsl_monte_plain_alloc . . . . . . . . . . . . . . . . . . . . 268 gsl_monte_plain_free. . . . . . . . . . . . . . . . . . . . . . 268 gsl_monte_plain_init. . . . . . . . . . . . . . . . . . . . . . 268

                                                gsl_monte_plain_integrate . . . . . . . . . . . . . . . . 268

gsl_monte_vegas_alloc . . . . . . . . . . . . . . . . . . . . 271 gsl_monte_vegas_free. . . . . . . . . . . . . . . . . . . . . . 271 gsl_monte_vegas_init. . . . . . . . . . . . . . . . . . . . . . 271

                                                gsl_monte_vegas_integrate . . . . . . . . . . . . . . . . 271

gsl_multifit_covar. . . . . . . . . . . . . . . . . . . . . . . . 381

                                                gsl_multifit_fdfsolver_alloc . . . . . . . . . . . . . 377
                                                gsl_multifit_fdfsolver_free . . . . . . . . . . . . . . 377
                                                gsl_multifit_fdfsolver_iterate . . . . . . . . . . . 378
                                                gsl_multifit_fdfsolver_lmder . . . . . . . . . . . . . 381
                                                gsl_multifit_fdfsolver_lmsder . . . . . . . . . . . . 380
                                                gsl_multifit_fdfsolver_name . . . . . . . . . . . . . . 377
                                                gsl_multifit_fdfsolver_position. . . . . . . . . . 379
                                                gsl_multifit_fdfsolver_set . . . . . . . . . . . . . . . 377
                                                gsl_multifit_fsolver_alloc . . . . . . . . . . . . . . . 376
                                                gsl_multifit_fsolver_free . . . . . . . . . . . . . . . . 377
                                                gsl_multifit_fsolver_iterate . . . . . . . . . . . . . 378
                                                gsl_multifit_fsolver_name . . . . . . . . . . . . . . . . 377
                                                gsl_multifit_fsolver_position . . . . . . . . . . . . 379
                                                gsl_multifit_fsolver_set . . . . . . . . . . . . . . . . . 377

gsl_multifit_gradient . . . . . . . . . . . . . . . . . . . . 380 gsl_multifit_linear. . . . . . . . . . . . . . . . . . . . . . . 369

                                                gsl_multifit_linear_alloc . . . . . . . . . . . . . . . . 368
                                                gsl_multifit_linear_est . . . . . . . . . . . . . . . . . . 369
                                                gsl_multifit_linear_free . . . . . . . . . . . . . . . . . 368
                                                gsl_multifit_linear_svd . . . . . . . . . . . . . . . . . . 369
                                                gsl_multifit_test_delta . . . . . . . . . . . . . . . . . . 379
                                                gsl_multifit_test_gradient . . . . . . . . . . . . . . . 379

gsl_multifit_wlinear. . . . . . . . . . . . . . . . . . . . . . 369

                                                gsl_multifit_wlinear_svd . . . . . . . . . . . . . . . . . 369
                                                gsl_multimin_fdfminimizer_alloc. . . . . . . . . . 356
                                                gsl_multimin_fdfminimizer_conjugate_fr . . 360
                                                gsl_multimin_fdfminimizer_conjugate_pr . . 360
                                                gsl_multimin_fdfminimizer_free . . . . . . . . . . . 356
                                                gsl_multimin_fdfminimizer_gradient . . . . . . 359
                                                gsl_multimin_fdfminimizer_iterate . . . . . . . 359
                                                gsl_multimin_fdfminimizer_minimum . . . . . . . 359
                                                gsl_multimin_fdfminimizer_name . . . . . . . . . . . 356
                                                gsl_multimin_fdfminimizer_restart . . . . . . . 359
                                                gsl_multimin_fdfminimizer_set . . . . . . . . . . . . 356
                                                gsl_multimin_fdfminimizer_steepest_descent

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

                                                gsl_multimin_fdfminimizer_vector_bfgs . . . 360
                                                gsl_multimin_fdfminimizer_x . . . . . . . . . . . . . . 359
                                                gsl_multimin_fminimizer_alloc . . . . . . . . . . . . 356
                                                gsl_multimin_fminimizer_free . . . . . . . . . . . . . 356
                                                gsl_multimin_fminimizer_iterate. . . . . . . . . . 359
                                                gsl_multimin_fminimizer_minimum. . . . . . . . . . 359
                                                gsl_multimin_fminimizer_name . . . . . . . . . . . . . 356
                                                gsl_multimin_fminimizer_nmsimplex . . . . . . . 361
                                                gsl_multimin_fminimizer_set . . . . . . . . . . . . . . 356
                                                gsl_multimin_fminimizer_size . . . . . . . . . . . . . 359

gsl_multimin_fminimizer_x . . . . . . . . . . . . . . . . 359
gsl_multimin_test_gradient . . . . . . . . . . . . . . . 359
gsl_multimin_test_size . . . . . . . . . . . . . . . . . . . 360
gsl_multiroot_fdfsolver_alloc . . . . . . . . . . . . 342
gsl_multiroot_fdfsolver_dx . . . . . . . . . . . . . . . 346
gsl_multiroot_fdfsolver_f . . . . . . . . . . . . . . . . 346
gsl_multiroot_fdfsolver_free . . . . . . . . . . . . . 342
gsl_multiroot_fdfsolver_gnewton. . . . . . . . . . 348
gsl_multiroot_fdfsolver_hybridj. . . . . . . . . . 347
gsl_multiroot_fdfsolver_hybridsj. . . . . . . . . 347
gsl_multiroot_fdfsolver_iterate. . . . . . . . . . 345
gsl_multiroot_fdfsolver_name . . . . . . . . . . . . . 342
gsl_multiroot_fdfsolver_newton . . . . . . . . . . . 348
gsl_multiroot_fdfsolver_root . . . . . . . . . . . . . 346
gsl_multiroot_fdfsolver_set . . . . . . . . . . . . . . 342
gsl_multiroot_fsolver_alloc . . . . . . . . . . . . . . 342
gsl_multiroot_fsolver_broyden . . . . . . . . . . . . 349
gsl_multiroot_fsolver_dnewton . . . . . . . . . . . . 348
gsl_multiroot_fsolver_dx . . . . . . . . . . . . . . . . . 346
gsl_multiroot_fsolver_f . . . . . . . . . . . . . . . . . . 346
gsl_multiroot_fsolver_free . . . . . . . . . . . . . . . 342
gsl_multiroot_fsolver_hybrid . . . . . . . . . . . . . 348
gsl_multiroot_fsolver_hybrids . . . . . . . . . . . . 348
gsl_multiroot_fsolver_iterate . . . . . . . . . . . . 345
gsl_multiroot_fsolver_name . . . . . . . . . . . . . . . 342
gsl_multiroot_fsolver_root . . . . . . . . . . . . . . . 346
gsl_multiroot_fsolver_set . . . . . . . . . . . . . . . . 342
gsl_multiroot_test_delta . . . . . . . . . . . . . . . . . 346
gsl_multiroot_test_residual . . . . . . . . . . . . . . 346

GSL_NAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 GSL_NEGINF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 gsl_ntuple_bookdata. . . . . . . . . . . . . . . . . . . . . . . 261 gsl_ntuple_close . . . . . . . . . . . . . . . . . . . . . . . . . . 261 gsl_ntuple_create . . . . . . . . . . . . . . . . . . . . . . . . . 260 gsl_ntuple_open . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 gsl_ntuple_project. . . . . . . . . . . . . . . . . . . . . . . . 261 gsl_ntuple_read . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 gsl_ntuple_write . . . . . . . . . . . . . . . . . . . . . . . . . . 261

gsl_odeiv_control_alloc . . . . . . . . . . . . . . . . . . 288
gsl_odeiv_control_free . . . . . . . . . . . . . . . . . . . 288
gsl_odeiv_control_hadjust . . . . . . . . . . . . . . . . 289
gsl_odeiv_control_init . . . . . . . . . . . . . . . . . . . 288
gsl_odeiv_control_name . . . . . . . . . . . . . . . . . . . 289
gsl_odeiv_control_scaled_new . . . . . . . . . . . . . 288
gsl_odeiv_control_standard_new . . . . . . . . . . . 287
gsl_odeiv_control_y_new . . . . . . . . . . . . . . . . . . 288
gsl_odeiv_control_yp_new . . . . . . . . . . . . . . . . . 288
gsl_odeiv_evolve_alloc . . . . . . . . . . . . . . . . . . . 289
gsl_odeiv_evolve_apply . . . . . . . . . . . . . . . . . . . 289

gsl_odeiv_evolve_free . . . . . . . . . . . . . . . . . . . . 290

gsl_odeiv_evolve_reset . . . . . . . . . . . . . . . . . . . 290

gsl_odeiv_step_alloc. . . . . . . . . . . . . . . . . . . . . . 286 gsl_odeiv_step_apply. . . . . . . . . . . . . . . . . . . . . . 286 gsl_odeiv_step_bsimp. . . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_free. . . . . . . . . . . . . . . . . . . . . . . 286 gsl_odeiv_step_gear1. . . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_gear2. . . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_name. . . . . . . . . . . . . . . . . . . . . . . 286 gsl_odeiv_step_order. . . . . . . . . . . . . . . . . . . . . . 286 gsl_odeiv_step_reset. . . . . . . . . . . . . . . . . . . . . . 286 gsl_odeiv_step_rk2. . . . . . . . . . . . . . . . . . . . . . . . 286 gsl_odeiv_step_rk2imp . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_rk4. . . . . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_rk4imp . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_rk8pd. . . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_rkck. . . . . . . . . . . . . . . . . . . . . . . 287 gsl_odeiv_step_rkf45. . . . . . . . . . . . . . . . . . . . . . 287 gsl_permutation_alloc . . . . . . . . . . . . . . . . . . . . . 93

                                                gsl_permutation_calloc . . . . . . . . . . . . . . . . . . . . 93
                                                gsl_permutation_canonical_cycles. . . . . . . . . . 97
                                                gsl_permutation_canonical_to_linear . . . . . . 97

gsl_permutation_data. . . . . . . . . . . . . . . . . . . . . . . 94

                                                gsl_permutation_fprintf . . . . . . . . . . . . . . . . . . . 96

gsl_permutation_fread . . . . . . . . . . . . . . . . . . . . . 95 gsl_permutation_free. . . . . . . . . . . . . . . . . . . . . . . 93

                                                gsl_permutation_fscanf . . . . . . . . . . . . . . . . . . . . 96
                                                gsl_permutation_fwrite . . . . . . . . . . . . . . . . . . . . 95

gsl_permutation_get. . . . . . . . . . . . . . . . . . . . . . . . 94 gsl_permutation_init. . . . . . . . . . . . . . . . . . . . . . . 93

                                                gsl_permutation_inverse . . . . . . . . . . . . . . . . . . . 94
                                                gsl_permutation_inversions . . . . . . . . . . . . . . . . 97
                                                gsl_permutation_linear_cycles . . . . . . . . . . . . . 97
                                                gsl_permutation_linear_to_canonical . . . . . . 96
                                                gsl_permutation_memcpy . . . . . . . . . . . . . . . . . . . . 93

gsl_permutation_mul. . . . . . . . . . . . . . . . . . . . . . . . 95 gsl_permutation_next. . . . . . . . . . . . . . . . . . . . . . . 94 gsl_permutation_prev. . . . . . . . . . . . . . . . . . . . . . . 94

                                                gsl_permutation_reverse . . . . . . . . . . . . . . . . . . . 94

gsl_permutation_size. . . . . . . . . . . . . . . . . . . . . . . 94 gsl_permutation_swap. . . . . . . . . . . . . . . . . . . . . . . 94 gsl_permutation_valid . . . . . . . . . . . . . . . . . . . . . 94 gsl_permute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 gsl_permute_inverse. . . . . . . . . . . . . . . . . . . . . . . . 95 gsl_permute_vector . . . . . . . . . . . . . . . . . . . . . . . . . 95

                                                gsl_permute_vector_inverse . . . . . . . . . . . . . . . . 95
                                                gsl_poly_complex_solve . . . . . . . . . . . . . . . . . . . . 29
                                                gsl_poly_complex_solve_cubic . . . . . . . . . . . . . . 28
                                                gsl_poly_complex_solve_quadratic. . . . . . . . . . 28
                                                gsl_poly_complex_workspace_alloc. . . . . . . . . . 29
                                                gsl_poly_complex_workspace_free . . . . . . . . . . . 29

gsl_poly_dd_eval . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 gsl_poly_dd_init . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 gsl_poly_dd_taylor . . . . . . . . . . . . . . . . . . . . . . . . . 27 gsl_poly_eval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 gsl_poly_solve_cubic. . . . . . . . . . . . . . . . . . . . . . . 28

                                                gsl_poly_solve_quadratic . . . . . . . . . . . . . . . . . . 27

GSL_POSINF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 gsl_pow_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_pow_int . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_qrng_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 gsl_qrng_clone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 gsl_qrng_free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 gsl_qrng_get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 gsl_qrng_init . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 gsl_qrng_memcpy . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 gsl_qrng_name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

gsl_qrng_niederreiter_2 . . . . . . . . . . . . . . . . . . 185

gsl_qrng_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 gsl_qrng_sobol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 gsl_qrng_state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 gsl_ran_bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . 219 gsl_ran_bernoulli_pdf . . . . . . . . . . . . . . . . . . . . 219 gsl_ran_beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 gsl_ran_beta_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . 208 gsl_ran_binomial . . . . . . . . . . . . . . . . . . . . . . . . . . 220 gsl_ran_binomial_pdf. . . . . . . . . . . . . . . . . . . . . . 220

gsl_ran_bivariate_gaussian . . . . . . . . . . . . . . . 192
gsl_ran_bivariate_gaussian_pdf . . . . . . . . . . . 192

gsl_ran_cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 gsl_ran_cauchy_pdf. . . . . . . . . . . . . . . . . . . . . . . . 196 gsl_ran_chisq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 gsl_ran_chisq_pdf . . . . . . . . . . . . . . . . . . . . . . . . . 205 gsl_ran_choose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 gsl_ran_dir_2d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

gsl_ran_dir_2d_trig_method . . . . . . . . . . . . . . . 211

gsl_ran_dir_3d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 gsl_ran_dir_nd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 gsl_ran_dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . 215

gsl_ran_dirichlet_lnpdf . . . . . . . . . . . . . . . . . . 215

gsl_ran_dirichlet_pdf . . . . . . . . . . . . . . . . . . . . 215 gsl_ran_discrete . . . . . . . . . . . . . . . . . . . . . . . . . . 216 gsl_ran_discrete_free . . . . . . . . . . . . . . . . . . . . 217 gsl_ran_discrete_pdf. . . . . . . . . . . . . . . . . . . . . . 217

gsl_ran_discrete_preproc . . . . . . . . . . . . . . . . . 216

gsl_ran_exponential. . . . . . . . . . . . . . . . . . . . . . . 193

gsl_ran_exponential_pdf . . . . . . . . . . . . . . . . . . 193

gsl_ran_exppow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 gsl_ran_exppow_pdf. . . . . . . . . . . . . . . . . . . . . . . . 195 gsl_ran_fdist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 gsl_ran_fdist_pdf . . . . . . . . . . . . . . . . . . . . . . . . . 206 gsl_ran_flat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 gsl_ran_flat_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . 203 gsl_ran_gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 gsl_ran_gamma_mt . . . . . . . . . . . . . . . . . . . . . . . . . . 202 gsl_ran_gamma_pdf . . . . . . . . . . . . . . . . . . . . . . . . . 202 gsl_ran_gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . 189 gsl_ran_gaussian_pdf. . . . . . . . . . . . . . . . . . . . . . 189

gsl_ran_gaussian_ratio_method . . . . . . . . . . . . 189

gsl_ran_gaussian_tail . . . . . . . . . . . . . . . . . . . . 191

gsl_ran_gaussian_tail_pdf . . . . . . . . . . . . . . . . 191
gsl_ran_gaussian_ziggurat . . . . . . . . . . . . . . . . 189

gsl_ran_geometric . . . . . . . . . . . . . . . . . . . . . . . . . 224 gsl_ran_geometric_pdf . . . . . . . . . . . . . . . . . . . . 224 gsl_ran_gumbel1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 gsl_ran_gumbel1_pdf. . . . . . . . . . . . . . . . . . . . . . . 213 gsl_ran_gumbel2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 gsl_ran_gumbel2_pdf. . . . . . . . . . . . . . . . . . . . . . . 214

gsl_ran_hypergeometric . . . . . . . . . . . . . . . . . . . 225
gsl_ran_hypergeometric_pdf . . . . . . . . . . . . . . . 225

gsl_ran_landau . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 gsl_ran_landau_pdf. . . . . . . . . . . . . . . . . . . . . . . . 199 gsl_ran_laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 gsl_ran_laplace_pdf. . . . . . . . . . . . . . . . . . . . . . . 194 gsl_ran_levy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 gsl_ran_levy_skew . . . . . . . . . . . . . . . . . . . . . . . . . 201 gsl_ran_logarithmic. . . . . . . . . . . . . . . . . . . . . . . 226

                                                gsl_ran_logarithmic_pdf . . . . . . . . . . . . . . . . . . 226

gsl_ran_logistic . . . . . . . . . . . . . . . . . . . . . . . . . . 209 gsl_ran_logistic_pdf. . . . . . . . . . . . . . . . . . . . . . 209 gsl_ran_lognormal . . . . . . . . . . . . . . . . . . . . . . . . . 204 gsl_ran_lognormal_pdf . . . . . . . . . . . . . . . . . . . . 204 gsl_ran_multinomial. . . . . . . . . . . . . . . . . . . . . . . 221

                                                gsl_ran_multinomial_lnpdf . . . . . . . . . . . . . . . . 221
                                                gsl_ran_multinomial_pdf . . . . . . . . . . . . . . . . . . 221
                                                gsl_ran_negative_binomial . . . . . . . . . . . . . . . . 222
                                                gsl_ran_negative_binomial_pdf . . . . . . . . . . . . 222

gsl_ran_pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 gsl_ran_pareto_pdf. . . . . . . . . . . . . . . . . . . . . . . . 210 gsl_ran_pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 gsl_ran_pascal_pdf. . . . . . . . . . . . . . . . . . . . . . . . 223 gsl_ran_poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 gsl_ran_poisson_pdf. . . . . . . . . . . . . . . . . . . . . . . 218 gsl_ran_rayleigh . . . . . . . . . . . . . . . . . . . . . . . . . . 197 gsl_ran_rayleigh_pdf. . . . . . . . . . . . . . . . . . . . . . 197 gsl_ran_rayleigh_tail . . . . . . . . . . . . . . . . . . . . 198

                                                gsl_ran_rayleigh_tail_pdf . . . . . . . . . . . . . . . . 198

gsl_ran_sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 gsl_ran_shuffle . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 gsl_ran_tdist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 gsl_ran_tdist_pdf . . . . . . . . . . . . . . . . . . . . . . . . . 207 gsl_ran_ugaussian . . . . . . . . . . . . . . . . . . . . . . . . . 189 gsl_ran_ugaussian_pdf . . . . . . . . . . . . . . . . . . . . 189

                                                gsl_ran_ugaussian_ratio_method . . . . . . . . . . . 189
                                                gsl_ran_ugaussian_tail . . . . . . . . . . . . . . . . . . . 191
                                                gsl_ran_ugaussian_tail_pdf . . . . . . . . . . . . . . . 191

gsl_ran_weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 gsl_ran_weibull_pdf. . . . . . . . . . . . . . . . . . . . . . . 212 GSL_REAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 gsl_rng_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 gsl_rng_borosh13 . . . . . . . . . . . . . . . . . . . . . . . . . . 179 gsl_rng_clone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 gsl_rng_cmrg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_coveyou . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 gsl_rng_env_setup . . . . . . . . . . . . . . . . . . . . . . . . . 170 gsl_rng_fishman18 . . . . . . . . . . . . . . . . . . . . . . . . . 179 gsl_rng_fishman20 . . . . . . . . . . . . . . . . . . . . . . . . . 179 gsl_rng_fishman2x . . . . . . . . . . . . . . . . . . . . . . . . . 180 gsl_rng_fread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 gsl_rng_free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 gsl_rng_fwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 gsl_rng_get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 gsl_rng_gfsr4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 gsl_rng_knuthran . . . . . . . . . . . . . . . . . . . . . . . . . . 180 gsl_rng_knuthran2 . . . . . . . . . . . . . . . . . . . . . . . . . 180 gsl_rng_lecuyer21 . . . . . . . . . . . . . . . . . . . . . . . . . 180 gsl_rng_max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 gsl_rng_memcpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 gsl_rng_min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 gsl_rng_minstd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 gsl_rng_mrg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 gsl_rng_mt19937 . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 gsl_rng_name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 gsl_rng_r250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 gsl_rng_rand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 gsl_rng_rand48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 gsl_rng_random_bsd. . . . . . . . . . . . . . . . . . . . . . . . 176 gsl_rng_random_glibc2 . . . . . . . . . . . . . . . . . . . . 176 gsl_rng_random_libc5. . . . . . . . . . . . . . . . . . . . . . 176 gsl_rng_randu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 gsl_rng_ranf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 gsl_rng_ranlux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_ranlux389 . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_ranlxd1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_ranlxd2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_ranlxs0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_ranlxs1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_ranlxs2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 gsl_rng_ranmar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 gsl_rng_set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 gsl_rng_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 gsl_rng_slatec . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 gsl_rng_state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 gsl_rng_taus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 gsl_rng_taus2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 gsl_rng_transputer. . . . . . . . . . . . . . . . . . . . . . . . 178 gsl_rng_tt800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 gsl_rng_types_setup. . . . . . . . . . . . . . . . . . . . . . . 170 gsl_rng_uni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 gsl_rng_uni32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 gsl_rng_uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 gsl_rng_uniform_int. . . . . . . . . . . . . . . . . . . . . . . 169 gsl_rng_uniform_pos. . . . . . . . . . . . . . . . . . . . . . . 169 gsl_rng_vax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 gsl_rng_waterman14. . . . . . . . . . . . . . . . . . . . . . . . 180 gsl_rng_zuf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

gsl_root_fdfsolver_alloc . . . . . . . . . . . . . . . . . 322
gsl_root_fdfsolver_free . . . . . . . . . . . . . . . . . . 323
gsl_root_fdfsolver_iterate . . . . . . . . . . . . . . . 325
gsl_root_fdfsolver_name . . . . . . . . . . . . . . . . . . 323
gsl_root_fdfsolver_newton . . . . . . . . . . . . . . . . 328
gsl_root_fdfsolver_root . . . . . . . . . . . . . . . . . . 326
gsl_root_fdfsolver_secant . . . . . . . . . . . . . . . . 328
gsl_root_fdfsolver_set . . . . . . . . . . . . . . . . . . . 322
gsl_root_fdfsolver_steffenson . . . . . . . . . . . . 329
gsl_root_fsolver_alloc . . . . . . . . . . . . . . . . . . . 322
gsl_root_fsolver_bisection . . . . . . . . . . . . . . . 327
gsl_root_fsolver_brent . . . . . . . . . . . . . . . . . . . 327
gsl_root_fsolver_falsepos . . . . . . . . . . . . . . . . 327

gsl_root_fsolver_free . . . . . . . . . . . . . . . . . . . . 323

gsl_root_fsolver_iterate . . . . . . . . . . . . . . . . . 325

gsl_root_fsolver_name . . . . . . . . . . . . . . . . . . . . 323 gsl_root_fsolver_root . . . . . . . . . . . . . . . . . . . . 326 gsl_root_fsolver_set. . . . . . . . . . . . . . . . . . . . . . 322

gsl_root_fsolver_x_lower . . . . . . . . . . . . . . . . . 326
gsl_root_fsolver_x_upper . . . . . . . . . . . . . . . . . 326

gsl_root_test_delta. . . . . . . . . . . . . . . . . . . . . . . 326

gsl_root_test_interval . . . . . . . . . . . . . . . . . . . 326
gsl_root_test_residual . . . . . . . . . . . . . . . . . . . 327

GSL_SET_COMPLEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 gsl_set_error_handler . . . . . . . . . . . . . . . . . . . . . 13

gsl_set_error_handler_off . . . . . . . . . . . . . . . . . 13

GSL_SET_IMAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 GSL_SET_REAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gsl_sf_airy_Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 gsl_sf_airy_Ai_deriv. . . . . . . . . . . . . . . . . . . . . . . 33

gsl_sf_airy_Ai_deriv_e . . . . . . . . . . . . . . . . . . . . 33
gsl_sf_airy_Ai_deriv_scaled . . . . . . . . . . . . . . . 33
gsl_sf_airy_Ai_deriv_scaled_e . . . . . . . . . . . . . 33

gsl_sf_airy_Ai_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 gsl_sf_airy_Ai_scaled . . . . . . . . . . . . . . . . . . . . . 33

                                                gsl_sf_airy_Ai_scaled_e . . . . . . . . . . . . . . . . . . . 33

gsl_sf_airy_Bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 gsl_sf_airy_Bi_deriv. . . . . . . . . . . . . . . . . . . . . . . 33

                                                gsl_sf_airy_Bi_deriv_e . . . . . . . . . . . . . . . . . . . . 33
                                                gsl_sf_airy_Bi_deriv_scaled . . . . . . . . . . . . . . . 33
                                                gsl_sf_airy_Bi_deriv_scaled_e . . . . . . . . . . . . . 33

gsl_sf_airy_Bi_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 gsl_sf_airy_Bi_scaled . . . . . . . . . . . . . . . . . . . . . 33

                                                gsl_sf_airy_Bi_scaled_e . . . . . . . . . . . . . . . . . . . 33

gsl_sf_airy_zero_Ai. . . . . . . . . . . . . . . . . . . . . . . . 33

                                                gsl_sf_airy_zero_Ai_deriv . . . . . . . . . . . . . . . . . 34
                                                gsl_sf_airy_zero_Ai_deriv_e . . . . . . . . . . . . . . . 34

gsl_sf_airy_zero_Ai_e . . . . . . . . . . . . . . . . . . . . . 33 gsl_sf_airy_zero_Bi. . . . . . . . . . . . . . . . . . . . . . . . 33

                                                gsl_sf_airy_zero_Bi_deriv . . . . . . . . . . . . . . . . . 34
                                                gsl_sf_airy_zero_Bi_deriv_e . . . . . . . . . . . . . . . 34

gsl_sf_airy_zero_Bi_e . . . . . . . . . . . . . . . . . . . . . 33

                                                gsl_sf_angle_restrict_pos . . . . . . . . . . . . . . . . . 66
                                                gsl_sf_angle_restrict_pos_e . . . . . . . . . . . . . . . 66
                                                gsl_sf_angle_restrict_symm . . . . . . . . . . . . . . . . 66
                                                gsl_sf_angle_restrict_symm_e . . . . . . . . . . . . . . 66

gsl_sf_atanint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 gsl_sf_atanint_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 gsl_sf_bessel_I0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 gsl_sf_bessel_I0_e . . . . . . . . . . . . . . . . . . . . . . . . . 35

                                                gsl_sf_bessel_i0_scaled . . . . . . . . . . . . . . . . . . . 38
                                                gsl_sf_bessel_I0_scaled . . . . . . . . . . . . . . . . . . . 35
                                                gsl_sf_bessel_i0_scaled_e . . . . . . . . . . . . . . . . . 38
                                                gsl_sf_bessel_I0_scaled_e . . . . . . . . . . . . . . . . . 35

gsl_sf_bessel_I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 gsl_sf_bessel_I1_e . . . . . . . . . . . . . . . . . . . . . . . . . 35

                                                gsl_sf_bessel_i1_scaled . . . . . . . . . . . . . . . . . . . 38
                                                gsl_sf_bessel_I1_scaled . . . . . . . . . . . . . . . . . . . 35
                                                gsl_sf_bessel_i1_scaled_e . . . . . . . . . . . . . . . . . 38
                                                gsl_sf_bessel_I1_scaled_e . . . . . . . . . . . . . . . . . 35
                                                gsl_sf_bessel_i2_scaled . . . . . . . . . . . . . . . . . . . 38
                                                gsl_sf_bessel_i2_scaled_e . . . . . . . . . . . . . . . . . 38
                                                gsl_sf_bessel_il_scaled . . . . . . . . . . . . . . . . . . . 39
                                                gsl_sf_bessel_il_scaled_array . . . . . . . . . . . . . 39
                                                gsl_sf_bessel_il_scaled_e . . . . . . . . . . . . . . . . . 39

gsl_sf_bessel_In . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

                                                gsl_sf_bessel_In_array . . . . . . . . . . . . . . . . . . . . 35

gsl_sf_bessel_In_e . . . . . . . . . . . . . . . . . . . . . . . . . 35

                                                gsl_sf_bessel_In_scaled . . . . . . . . . . . . . . . . . . . 36
                                                gsl_sf_bessel_In_scaled_array . . . . . . . . . . . . . 36
                                                gsl_sf_bessel_In_scaled_e . . . . . . . . . . . . . . . . . 36

gsl_sf_bessel_Inu . . . . . . . . . . . . . . . . . . . . . . . . . . 40 gsl_sf_bessel_Inu_e. . . . . . . . . . . . . . . . . . . . . . . . 40

                                                gsl_sf_bessel_Inu_scaled . . . . . . . . . . . . . . . . . . 40
                                                gsl_sf_bessel_Inu_scaled_e . . . . . . . . . . . . . . . . 40

gsl_sf_bessel_j0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 gsl_sf_bessel_J0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 gsl_sf_bessel_j0_e . . . . . . . . . . . . . . . . . . . . . . . . . 37 gsl_sf_bessel_J0_e . . . . . . . . . . . . . . . . . . . . . . . . . 34 gsl_sf_bessel_j1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 gsl_sf_bessel_J1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 gsl_sf_bessel_j1_e . . . . . . . . . . . . . . . . . . . . . . . . . 37 gsl_sf_bessel_J1_e . . . . . . . . . . . . . . . . . . . . . . . . . 34 gsl_sf_bessel_j2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 gsl_sf_bessel_j2_e . . . . . . . . . . . . . . . . . . . . . . . . . 37 gsl_sf_bessel_jl . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

gsl_sf_bessel_jl_array . . . . . . . . . . . . . . . . . . . . 37

gsl_sf_bessel_jl_e . . . . . . . . . . . . . . . . . . . . . . . . . 37

gsl_sf_bessel_jl_steed_array . . . . . . . . . . . . . . 37

gsl_sf_bessel_Jn . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

gsl_sf_bessel_Jn_array . . . . . . . . . . . . . . . . . . . . 34

gsl_sf_bessel_Jn_e . . . . . . . . . . . . . . . . . . . . . . . . . 34 gsl_sf_bessel_Jnu . . . . . . . . . . . . . . . . . . . . . . . . . . 39 gsl_sf_bessel_Jnu_e. . . . . . . . . . . . . . . . . . . . . . . . 39 gsl_sf_bessel_K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 gsl_sf_bessel_K0_e . . . . . . . . . . . . . . . . . . . . . . . . . 36

gsl_sf_bessel_k0_scaled . . . . . . . . . . . . . . . . . . . 39
gsl_sf_bessel_K0_scaled . . . . . . . . . . . . . . . . . . . 36
gsl_sf_bessel_k0_scaled_e . . . . . . . . . . . . . . . . . 39
gsl_sf_bessel_K0_scaled_e . . . . . . . . . . . . . . . . . 36

gsl_sf_bessel_K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 gsl_sf_bessel_K1_e . . . . . . . . . . . . . . . . . . . . . . . . . 36

gsl_sf_bessel_k1_scaled . . . . . . . . . . . . . . . . . . . 39
gsl_sf_bessel_K1_scaled . . . . . . . . . . . . . . . . . . . 36
gsl_sf_bessel_k1_scaled_e . . . . . . . . . . . . . . . . . 39
gsl_sf_bessel_K1_scaled_e . . . . . . . . . . . . . . . . . 36
gsl_sf_bessel_k2_scaled . . . . . . . . . . . . . . . . . . . 39
gsl_sf_bessel_k2_scaled_e . . . . . . . . . . . . . . . . . 39
gsl_sf_bessel_kl_scaled . . . . . . . . . . . . . . . . . . . 39
gsl_sf_bessel_kl_scaled_array . . . . . . . . . . . . . 39
gsl_sf_bessel_kl_scaled_e . . . . . . . . . . . . . . . . . 39

gsl_sf_bessel_Kn . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

gsl_sf_bessel_Kn_array . . . . . . . . . . . . . . . . . . . . 36

gsl_sf_bessel_Kn_e . . . . . . . . . . . . . . . . . . . . . . . . . 36

gsl_sf_bessel_Kn_scaled . . . . . . . . . . . . . . . . . . . 37
gsl_sf_bessel_Kn_scaled_array . . . . . . . . . . . . . 37
gsl_sf_bessel_Kn_scaled_e . . . . . . . . . . . . . . . . . 37

gsl_sf_bessel_Knu . . . . . . . . . . . . . . . . . . . . . . . . . . 40 gsl_sf_bessel_Knu_e. . . . . . . . . . . . . . . . . . . . . . . . 40

gsl_sf_bessel_Knu_scaled . . . . . . . . . . . . . . . . . . 40
gsl_sf_bessel_Knu_scaled_e . . . . . . . . . . . . . . . . 40

gsl_sf_bessel_lnKnu. . . . . . . . . . . . . . . . . . . . . . . . 40 gsl_sf_bessel_lnKnu_e . . . . . . . . . . . . . . . . . . . . . 40

gsl_sf_bessel_sequence_Jnu_e . . . . . . . . . . . . . . 40

gsl_sf_bessel_y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 gsl_sf_bessel_Y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 gsl_sf_bessel_y0_e . . . . . . . . . . . . . . . . . . . . . . . . . 38 gsl_sf_bessel_Y0_e . . . . . . . . . . . . . . . . . . . . . . . . . 34 gsl_sf_bessel_y1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 gsl_sf_bessel_Y1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 gsl_sf_bessel_y1_e . . . . . . . . . . . . . . . . . . . . . . . . . 38 gsl_sf_bessel_Y1_e . . . . . . . . . . . . . . . . . . . . . . . . . 35 gsl_sf_bessel_y2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 gsl_sf_bessel_y2_e . . . . . . . . . . . . . . . . . . . . . . . . . 38 gsl_sf_bessel_yl . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

gsl_sf_bessel_yl_array . . . . . . . . . . . . . . . . . . . . 38

gsl_sf_bessel_yl_e . . . . . . . . . . . . . . . . . . . . . . . . . 38 gsl_sf_bessel_Yn . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

gsl_sf_bessel_Yn_array . . . . . . . . . . . . . . . . . . . . 35

gsl_sf_bessel_Yn_e . . . . . . . . . . . . . . . . . . . . . . . . . 35 gsl_sf_bessel_Ynu . . . . . . . . . . . . . . . . . . . . . . . . . . 40 gsl_sf_bessel_Ynu_e. . . . . . . . . . . . . . . . . . . . . . . . 40 gsl_sf_bessel_zero_J0 . . . . . . . . . . . . . . . . . . . . . 41

gsl_sf_bessel_zero_J0_e . . . . . . . . . . . . . . . . . . . 41

gsl_sf_bessel_zero_J1 . . . . . . . . . . . . . . . . . . . . . 41

                                                gsl_sf_bessel_zero_J1_e . . . . . . . . . . . . . . . . . . . 41
                                                gsl_sf_bessel_zero_Jnu . . . . . . . . . . . . . . . . . . . . 41
                                                gsl_sf_bessel_zero_Jnu_e . . . . . . . . . . . . . . . . . . 41

gsl_sf_beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_beta_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_beta_inc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_beta_inc_e . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_Chi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_Chi_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_choose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_choose_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_Ci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 gsl_sf_Ci_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 gsl_sf_clausen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 gsl_sf_clausen_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 gsl_sf_complex_cos_e. . . . . . . . . . . . . . . . . . . . . . . 66

                                                gsl_sf_complex_dilog_e . . . . . . . . . . . . . . . . . . . . 45

gsl_sf_complex_log_e. . . . . . . . . . . . . . . . . . . . . . . 63

                                                gsl_sf_complex_logsin_e . . . . . . . . . . . . . . . . . . . 66

gsl_sf_complex_sin_e. . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_conicalP_0 . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gsl_sf_conicalP_0_e. . . . . . . . . . . . . . . . . . . . . . . . 62 gsl_sf_conicalP_1 . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gsl_sf_conicalP_1_e. . . . . . . . . . . . . . . . . . . . . . . . 62

                                                gsl_sf_conicalP_cyl_reg . . . . . . . . . . . . . . . . . . . 62
                                                gsl_sf_conicalP_cyl_reg_e . . . . . . . . . . . . . . . . . 62

gsl_sf_conicalP_half. . . . . . . . . . . . . . . . . . . . . . . 61

                                                gsl_sf_conicalP_half_e . . . . . . . . . . . . . . . . . . . . 61

gsl_sf_conicalP_mhalf . . . . . . . . . . . . . . . . . . . . . 61

                                                gsl_sf_conicalP_mhalf_e . . . . . . . . . . . . . . . . . . . 61
                                                gsl_sf_conicalP_sph_reg . . . . . . . . . . . . . . . . . . . 62
                                                gsl_sf_conicalP_sph_reg_e . . . . . . . . . . . . . . . . . 62

gsl_sf_cos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_cos_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_cos_err_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

                                                gsl_sf_coulomb_CL_array . . . . . . . . . . . . . . . . . . . 43

gsl_sf_coulomb_CL_e. . . . . . . . . . . . . . . . . . . . . . . . 43

                                                gsl_sf_coulomb_wave_F_array . . . . . . . . . . . . . . . 42
                                                gsl_sf_coulomb_wave_FG_array . . . . . . . . . . . . . . 42
                                                gsl_sf_coulomb_wave_FG_e . . . . . . . . . . . . . . . . . . 42
                                                gsl_sf_coulomb_wave_FGp_array . . . . . . . . . . . . . 43
                                                gsl_sf_coulomb_wave_sphF_array . . . . . . . . . . . . 43

gsl_sf_coupling_3j . . . . . . . . . . . . . . . . . . . . . . . . . 43 gsl_sf_coupling_3j_e. . . . . . . . . . . . . . . . . . . . . . . 43 gsl_sf_coupling_6j . . . . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_coupling_6j_e. . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_coupling_9j . . . . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_coupling_9j_e. . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_dawson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_dawson_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_debye_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_debye_1_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 gsl_sf_debye_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_2_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_3_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_4_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_5_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_debye_6_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_dilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_dilog_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 gsl_sf_doublefact . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_doublefact_e. . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_ellint_D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_D_e . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_E_e . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_Ecomp. . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_Ecomp_e . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_F_e . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_Kcomp. . . . . . . . . . . . . . . . . . . . . . . . 46 gsl_sf_ellint_Kcomp_e . . . . . . . . . . . . . . . . . . . . . 46 gsl_sf_ellint_P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_P_e . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_RC . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_RC_e . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_RD . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_RD_e . . . . . . . . . . . . . . . . . . . . . . . . . 47 gsl_sf_ellint_RF . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_ellint_RF_e . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_ellint_RJ . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_ellint_RJ_e . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_elljac_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_erf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_erf_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_erf_Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_erf_Q_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_erf_Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_erf_Z_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_erfc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_erfc_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_eta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 gsl_sf_eta_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 gsl_sf_eta_int . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 gsl_sf_eta_int_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 gsl_sf_exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_exp_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_exp_e10_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_exp_err_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exp_err_e10_e. . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exp_mult . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_exp_mult_e . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_exp_mult_e10_e . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_exp_mult_err_e . . . . . . . . . . . . . . . . . . . . . 50

gsl_sf_exp_mult_err_e10_e . . . . . . . . . . . . . . . . . 50

gsl_sf_expint_3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expint_3_e . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expint_E1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expint_E1_e . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expint_E2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expint_E2_e . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expint_Ei . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expint_Ei_e . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_expm1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_expm1_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exprel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exprel_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exprel_2_e . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exprel_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exprel_n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_exprel_n_e . . . . . . . . . . . . . . . . . . . . . . . . . . 50 gsl_sf_fact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_fact_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_fermi_dirac_0. . . . . . . . . . . . . . . . . . . . . . . 52

                                                gsl_sf_fermi_dirac_0_e . . . . . . . . . . . . . . . . . . . . 52

gsl_sf_fermi_dirac_1. . . . . . . . . . . . . . . . . . . . . . . 52

                                                gsl_sf_fermi_dirac_1_e . . . . . . . . . . . . . . . . . . . . 52

gsl_sf_fermi_dirac_2. . . . . . . . . . . . . . . . . . . . . . . 52

                                                gsl_sf_fermi_dirac_2_e . . . . . . . . . . . . . . . . . . . . 52
                                                gsl_sf_fermi_dirac_3half . . . . . . . . . . . . . . . . . . 53
                                                gsl_sf_fermi_dirac_3half_e . . . . . . . . . . . . . . . . 53
                                                gsl_sf_fermi_dirac_half . . . . . . . . . . . . . . . . . . . 53
                                                gsl_sf_fermi_dirac_half_e . . . . . . . . . . . . . . . . . 53
                                                gsl_sf_fermi_dirac_inc_0 . . . . . . . . . . . . . . . . . . 53
                                                gsl_sf_fermi_dirac_inc_0_e . . . . . . . . . . . . . . . . 53
                                                gsl_sf_fermi_dirac_int . . . . . . . . . . . . . . . . . . . . 52
                                                gsl_sf_fermi_dirac_int_e . . . . . . . . . . . . . . . . . . 52

gsl_sf_fermi_dirac_m1 . . . . . . . . . . . . . . . . . . . . . 52

                                                gsl_sf_fermi_dirac_m1_e . . . . . . . . . . . . . . . . . . . 52
                                                gsl_sf_fermi_dirac_mhalf . . . . . . . . . . . . . . . . . . 53
                                                gsl_sf_fermi_dirac_mhalf_e . . . . . . . . . . . . . . . . 53

gsl_sf_gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 gsl_sf_gamma_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 gsl_sf_gamma_inc . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_gamma_inc_e . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_gamma_inc_P . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_gamma_inc_P_e. . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_gamma_inc_Q . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_gamma_inc_Q_e. . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_gammainv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_gammainv_e . . . . . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_gammastar . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_gammastar_e . . . . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_gegenpoly_1 . . . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_gegenpoly_1_e. . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_gegenpoly_2 . . . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_gegenpoly_2_e. . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_gegenpoly_3 . . . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_gegenpoly_3_e. . . . . . . . . . . . . . . . . . . . . . . 57

                                                gsl_sf_gegenpoly_array . . . . . . . . . . . . . . . . . . . . 57

gsl_sf_gegenpoly_n . . . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_gegenpoly_n_e. . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_hazard_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 gsl_sf_hydrogenicR . . . . . . . . . . . . . . . . . . . . . . . . . 42 gsl_sf_hydrogenicR_1. . . . . . . . . . . . . . . . . . . . . . . 41

                                                gsl_sf_hydrogenicR_1_e . . . . . . . . . . . . . . . . . . . . 41

gsl_sf_hydrogenicR_e. . . . . . . . . . . . . . . . . . . . . . . 42 gsl_sf_hyperg_0F1 . . . . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_hyperg_0F1_e. . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_hyperg_1F1 . . . . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_hyperg_1F1_e. . . . . . . . . . . . . . . . . . . . . . . . 57 gsl_sf_hyperg_1F1_int . . . . . . . . . . . . . . . . . . . . . 57

                                                gsl_sf_hyperg_1F1_int_e . . . . . . . . . . . . . . . . . . . 57

gsl_sf_hyperg_2F0 . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_hyperg_2F0_e. . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_hyperg_2F1 . . . . . . . . . . . . . . . . . . . . . . . . . . 58

                                                gsl_sf_hyperg_2F1_conj . . . . . . . . . . . . . . . . . . . . 58
                                                gsl_sf_hyperg_2F1_conj_e . . . . . . . . . . . . . . . . . . 58

gsl_sf_hyperg_2F1_conj_renorm . . . . . . . . . . . . . 58
gsl_sf_hyperg_2F1_conj_renorm_e . . . . . . . . . . . 58

gsl_sf_hyperg_2F1_e. . . . . . . . . . . . . . . . . . . . . . . . 58

gsl_sf_hyperg_2F1_renorm . . . . . . . . . . . . . . . . . . 58
gsl_sf_hyperg_2F1_renorm_e . . . . . . . . . . . . . . . . 58

gsl_sf_hyperg_U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 gsl_sf_hyperg_U_e . . . . . . . . . . . . . . . . . . . . . . . . . . 58 gsl_sf_hyperg_U_e10_e . . . . . . . . . . . . . . . . . . . . . 58 gsl_sf_hyperg_U_int. . . . . . . . . . . . . . . . . . . . . . . . 58 gsl_sf_hyperg_U_int_e . . . . . . . . . . . . . . . . . . . . . 58

gsl_sf_hyperg_U_int_e10_e . . . . . . . . . . . . . . . . . 58

gsl_sf_hypot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_hypot_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_hzeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_hzeta_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_laguerre_1 . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_laguerre_1_e. . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_laguerre_2 . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_laguerre_2_e. . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_laguerre_3 . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_laguerre_3_e. . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_laguerre_n . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_laguerre_n_e. . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_lambert_W0 . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_lambert_W0_e. . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_lambert_Wm1 . . . . . . . . . . . . . . . . . . . . . . . . . 59 gsl_sf_lambert_Wm1_e. . . . . . . . . . . . . . . . . . . . . . . 59

gsl_sf_legendre_array_size . . . . . . . . . . . . . . . . 61

gsl_sf_legendre_H3d. . . . . . . . . . . . . . . . . . . . . . . . 62 gsl_sf_legendre_H3d_0 . . . . . . . . . . . . . . . . . . . . . 62

gsl_sf_legendre_H3d_0_e . . . . . . . . . . . . . . . . . . . 62

gsl_sf_legendre_H3d_1 . . . . . . . . . . . . . . . . . . . . . 62

gsl_sf_legendre_H3d_1_e . . . . . . . . . . . . . . . . . . . 62
gsl_sf_legendre_H3d_array . . . . . . . . . . . . . . . . . 63

gsl_sf_legendre_H3d_e . . . . . . . . . . . . . . . . . . . . . 62 gsl_sf_legendre_P1 . . . . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_P1_e. . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_P2 . . . . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_P2_e. . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_P3 . . . . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_P3_e. . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_Pl . . . . . . . . . . . . . . . . . . . . . . . . . 60

gsl_sf_legendre_Pl_array . . . . . . . . . . . . . . . . . . 60
gsl_sf_legendre_Pl_deriv_array . . . . . . . . . . . . 60

gsl_sf_legendre_Pl_e. . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_Plm. . . . . . . . . . . . . . . . . . . . . . . . 61

gsl_sf_legendre_Plm_array . . . . . . . . . . . . . . . . . 61
gsl_sf_legendre_Plm_deriv_array . . . . . . . . . . . 61

gsl_sf_legendre_Plm_e . . . . . . . . . . . . . . . . . . . . . 61 gsl_sf_legendre_Q0 . . . . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_Q0_e. . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_Q1 . . . . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_Q1_e. . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_Ql . . . . . . . . . . . . . . . . . . . . . . . . . 60 gsl_sf_legendre_Ql_e. . . . . . . . . . . . . . . . . . . . . . . 60

gsl_sf_legendre_sphPlm . . . . . . . . . . . . . . . . . . . . 61
gsl_sf_legendre_sphPlm_array . . . . . . . . . . . . . . 61
gsl_sf_legendre_sphPlm_deriv_array . . . . . . . 61
gsl_sf_legendre_sphPlm_e . . . . . . . . . . . . . . . . . . 61

gsl_sf_lnbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_lnbeta_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_lnchoose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lnchoose_e . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lncosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_lncosh_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_lndoublefact. . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lndoublefact_e . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lnfact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lnfact_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lngamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

                                                gsl_sf_lngamma_complex_e . . . . . . . . . . . . . . . . . . 54

gsl_sf_lngamma_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_lngamma_sgn_e. . . . . . . . . . . . . . . . . . . . . . . 54 gsl_sf_lnpoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lnpoch_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lnpoch_sgn_e. . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_lnsinh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_lnsinh_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_log_1plusx . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_log_1plusx_e. . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_log_1plusx_mx. . . . . . . . . . . . . . . . . . . . . . . 63

                                                gsl_sf_log_1plusx_mx_e . . . . . . . . . . . . . . . . . . . . 63

gsl_sf_log_abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_log_abs_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_log_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_log_erfc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_log_erfc_e . . . . . . . . . . . . . . . . . . . . . . . . . . 48 gsl_sf_multiply_e . . . . . . . . . . . . . . . . . . . . . . . . . . 46 gsl_sf_multiply_err_e . . . . . . . . . . . . . . . . . . . . . 46 gsl_sf_poch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_poch_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_pochrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_pochrel_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gsl_sf_polar_to_rect. . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_pow_int . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_pow_int_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gsl_sf_psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_1_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_1_int . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_1_int_e . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_1piy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_1piy_e . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_int . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_int_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_psi_n_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 gsl_sf_rect_to_polar. . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_Shi_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 gsl_sf_Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 gsl_sf_Si_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 gsl_sf_sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_sin_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_sin_err_e . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_sinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_sinc_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 gsl_sf_synchrotron_1. . . . . . . . . . . . . . . . . . . . . . . 65

                                                gsl_sf_synchrotron_1_e . . . . . . . . . . . . . . . . . . . . 65

gsl_sf_synchrotron_2. . . . . . . . . . . . . . . . . . . . . . . 65

gsl_sf_synchrotron_2_e . . . . . . . . . . . . . . . . . . . . 65

gsl_sf_taylorcoeff . . . . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_taylorcoeff_e. . . . . . . . . . . . . . . . . . . . . . . 55 gsl_sf_transport_2 . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_transport_2_e. . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_transport_3 . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_transport_3_e. . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_transport_4 . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_transport_4_e. . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_transport_5 . . . . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_transport_5_e. . . . . . . . . . . . . . . . . . . . . . . 65 gsl_sf_zeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_zeta_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_zeta_int . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_zeta_int_e . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_zetam1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_zetam1_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_zetam1_int . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gsl_sf_zetam1_int_e. . . . . . . . . . . . . . . . . . . . . . . . 67 GSL_SIGN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gsl_siman_solve . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 gsl_sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 gsl_sort_index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 gsl_sort_largest . . . . . . . . . . . . . . . . . . . . . . . . . . 106

gsl_sort_largest_index . . . . . . . . . . . . . . . . . . . 106

gsl_sort_smallest . . . . . . . . . . . . . . . . . . . . . . . . . 106

gsl_sort_smallest_index . . . . . . . . . . . . . . . . . . 106

gsl_sort_vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 gsl_sort_vector_index . . . . . . . . . . . . . . . . . . . . 105

gsl_sort_vector_largest . . . . . . . . . . . . . . . . . . 106
gsl_sort_vector_largest_index . . . . . . . . . . . . 106
gsl_sort_vector_smallest . . . . . . . . . . . . . . . . . 106
gsl_sort_vector_smallest_index . . . . . . . . . . . 106

gsl_spline_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . 297 gsl_spline_eval . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 gsl_spline_eval_deriv . . . . . . . . . . . . . . . . . . . . 297

gsl_spline_eval_deriv_e . . . . . . . . . . . . . . . . . . 297
gsl_spline_eval_deriv2 . . . . . . . . . . . . . . . . . . . 297
gsl_spline_eval_deriv2_e . . . . . . . . . . . . . . . . . 297

gsl_spline_eval_e . . . . . . . . . . . . . . . . . . . . . . . . . 297 gsl_spline_eval_integ . . . . . . . . . . . . . . . . . . . . 297

gsl_spline_eval_integ_e . . . . . . . . . . . . . . . . . . 297

gsl_spline_free . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 gsl_spline_init . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 gsl_spline_min_size. . . . . . . . . . . . . . . . . . . . . . . 297 gsl_spline_name . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 gsl_stats_absdev . . . . . . . . . . . . . . . . . . . . . . . . . . 233 gsl_stats_absdev_m. . . . . . . . . . . . . . . . . . . . . . . . 233 gsl_stats_covariance. . . . . . . . . . . . . . . . . . . . . . 235

gsl_stats_covariance_m . . . . . . . . . . . . . . . . . . . 235

gsl_stats_kurtosis. . . . . . . . . . . . . . . . . . . . . . . . 234

gsl_stats_kurtosis_m_sd . . . . . . . . . . . . . . . . . . 234
gsl_stats_lag1_autocorrelation . . . . . . . . . . . 235
gsl_stats_lag1_autocorrelation_m. . . . . . . . . 235

gsl_stats_max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 gsl_stats_max_index. . . . . . . . . . . . . . . . . . . . . . . 238 gsl_stats_mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

gsl_stats_median_from_sorted_data . . . . . . . 238

gsl_stats_min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 gsl_stats_min_index. . . . . . . . . . . . . . . . . . . . . . . 238 gsl_stats_minmax . . . . . . . . . . . . . . . . . . . . . . . . . . 238

                                                gsl_stats_minmax_index . . . . . . . . . . . . . . . . . . . 238
                                                gsl_stats_quantile_from_sorted_data . . . . . 238

gsl_stats_sd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 gsl_stats_sd_m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

                                                gsl_stats_sd_with_fixed_mean . . . . . . . . . . . . . 233

gsl_stats_skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 gsl_stats_skew_m_sd. . . . . . . . . . . . . . . . . . . . . . . 234 gsl_stats_variance. . . . . . . . . . . . . . . . . . . . . . . . 232 gsl_stats_variance_m. . . . . . . . . . . . . . . . . . . . . . 232

                                                gsl_stats_variance_with_fixed_mean . . . . . . 233

gsl_stats_wabsdev . . . . . . . . . . . . . . . . . . . . . . . . . 236 gsl_stats_wabsdev_m. . . . . . . . . . . . . . . . . . . . . . . 237 gsl_stats_wkurtosis. . . . . . . . . . . . . . . . . . . . . . . 237

                                                gsl_stats_wkurtosis_m_sd . . . . . . . . . . . . . . . . . 237

gsl_stats_wmean . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 gsl_stats_wsd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 gsl_stats_wsd_m . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

                                                gsl_stats_wsd_with_fixed_mean . . . . . . . . . . . . 236

gsl_stats_wskew . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 gsl_stats_wskew_m_sd. . . . . . . . . . . . . . . . . . . . . . 237 gsl_stats_wvariance. . . . . . . . . . . . . . . . . . . . . . . 236 gsl_stats_wvariance_m . . . . . . . . . . . . . . . . . . . . 236

                                                gsl_stats_wvariance_with_fixed_mean . . . . . 236

gsl_strerror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 gsl_sum_levin_u_accel . . . . . . . . . . . . . . . . . . . . 308 gsl_sum_levin_u_alloc . . . . . . . . . . . . . . . . . . . . 308 gsl_sum_levin_u_free. . . . . . . . . . . . . . . . . . . . . . 308

                                                gsl_sum_levin_utrunc_accel . . . . . . . . . . . . . . . 309
                                                gsl_sum_levin_utrunc_alloc . . . . . . . . . . . . . . . 309
                                                gsl_sum_levin_utrunc_free . . . . . . . . . . . . . . . . 309

gsl_vector_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

                                                gsl_vector_add_constant . . . . . . . . . . . . . . . . . . . 78

gsl_vector_alloc . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 gsl_vector_calloc . . . . . . . . . . . . . . . . . . . . . . . . . . 73

                                                gsl_vector_complex_const_imag . . . . . . . . . . . . . 76
                                                gsl_vector_complex_const_real . . . . . . . . . . . . . 76
                                                gsl_vector_complex_imag . . . . . . . . . . . . . . . . . . . 76
                                                gsl_vector_complex_real . . . . . . . . . . . . . . . . . . . 76

gsl_vector_const_ptr. . . . . . . . . . . . . . . . . . . . . . . 74

                                                gsl_vector_const_subvector . . . . . . . . . . . . . . . . 75
                                                gsl_vector_const_subvector_with_stride . . . 75
                                                gsl_vector_const_view_array . . . . . . . . . . . . . . . 76
                                                gsl_vector_const_view_array_with_stride . . 77

gsl_vector_div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_fprintf . . . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_fread . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_free . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 gsl_vector_fscanf . . . . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_fwrite . . . . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 gsl_vector_isnull . . . . . . . . . . . . . . . . . . . . . . . . . . 79 gsl_vector_max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_max_index. . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_memcpy . . . . . . . . . . . . . . . . . . . . . . . . . . 77 gsl_vector_min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_min_index. . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_minmax . . . . . . . . . . . . . . . . . . . . . . . . . . 78

                                                gsl_vector_minmax_index . . . . . . . . . . . . . . . . . . . 78

gsl_vector_mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_ptr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_reverse . . . . . . . . . . . . . . . . . . . . . . . . . 77 gsl_vector_scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_set_all . . . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_set_basis. . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_set_zero. . . . . . . . . . . . . . . . . . . . . . . . 74 gsl_vector_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 gsl_vector_subvector. . . . . . . . . . . . . . . . . . . . . . . 75

gsl_vector_subvector_with_stride. . . . . . . . . . 75

gsl_vector_swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

gsl_vector_swap_elements . . . . . . . . . . . . . . . . . . 77

gsl_vector_view_array . . . . . . . . . . . . . . . . . . . . . 76

gsl_vector_view_array_with_stride. . . . . . . . . 77

gsl_wavelet_alloc . . . . . . . . . . . . . . . . . . . . . . . . . 312 gsl_wavelet_bspline. . . . . . . . . . . . . . . . . . . . . . . 312

gsl_wavelet_bspline_centered . . . . . . . . . . . . . 312
gsl_wavelet_daubechies . . . . . . . . . . . . . . . . . . . 312
gsl_wavelet_daubechies_centered. . . . . . . . . . 312

gsl_wavelet_free . . . . . . . . . . . . . . . . . . . . . . . . . . 313 gsl_wavelet_haar . . . . . . . . . . . . . . . . . . . . . . . . . . 312

gsl_wavelet_haar_centered . . . . . . . . . . . . . . . . 312

gsl_wavelet_name . . . . . . . . . . . . . . . . . . . . . . . . . . 313 gsl_wavelet_transform . . . . . . . . . . . . . . . . . . . . 313

                                                gsl_wavelet_transform_forward . . . . . . . . . . . . 313
                                                gsl_wavelet_transform_inverse . . . . . . . . . . . . 313
                                                gsl_wavelet_workspace_alloc . . . . . . . . . . . . . . 313
                                                gsl_wavelet_workspace_free . . . . . . . . . . . . . . . 313
                                                gsl_wavelet2d_nstransform . . . . . . . . . . . . . . . . 315
                                                gsl_wavelet2d_nstransform_forward . . . . . . . 315
                                                gsl_wavelet2d_nstransform_inverse . . . . . . . 315
                                                gsl_wavelet2d_nstransform_matrix. . . . . . . . . 315
                                                gsl_wavelet2d_nstransform_matrix_forward

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

                                                gsl_wavelet2d_nstransform_matrix_inverse

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

                                                gsl_wavelet2d_transform . . . . . . . . . . . . . . . . . . 314
                                                gsl_wavelet2d_transform_forward. . . . . . . . . . 314
                                                gsl_wavelet2d_transform_inverse. . . . . . . . . . 314
                                                gsl_wavelet2d_transform_matrix . . . . . . . . . . . 314
                                                gsl_wavelet2d_transform_matrix_forward . . 314
                                                gsl_wavelet2d_transform_matrix_inverse . . 315

Appendix D: Variable Index 450

Variable Index

A

alpha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270, 272

C

chisq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

D

dither. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

E

estimate_frac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

I

iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

M

min_calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

min_calls_per_bisection . . . . . . . . . . . . . . . . . . 269

mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

O

ostream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

R

result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

S

sigma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

V

verbose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

Appendix D: Type Index 451

Type Index

gsl_error_handler_t. . . . . . . . . . . . . . . . . . . . . . . . 12

gsl_fft_complex_wavetable . . . . . . . . . . . . . . . . 146

gsl_function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 gsl_function_fdf . . . . . . . . . . . . . . . . . . . . . . . . . . 324 gsl_histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 gsl_histogram_pdf . . . . . . . . . . . . . . . . . . . . . . . . . 248 gsl_histogram2d . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 gsl_histogram2d_pdf. . . . . . . . . . . . . . . . . . . . . . . 256 gsl_monte_function. . . . . . . . . . . . . . . . . . . . . . . . 266 gsl_multifit_function . . . . . . . . . . . . . . . . . . . . 377

gsl_multifit_function_fdf . . . . . . . . . . . . . . . . 378

gsl_multimin_function . . . . . . . . . . . . . . . . . . . . 357

                                                gsl_multimin_function_fdf . . . . . . . . . . . . . . . . 357
                                                gsl_multiroot_function . . . . . . . . . . . . . . . . . . . 343
                                                gsl_multiroot_function_fdf . . . . . . . . . . . . . . . 344

gsl_odeiv_system . . . . . . . . . . . . . . . . . . . . . . . . . . 285

                                                gsl_siman_copy_construct_t . . . . . . . . . . . . . . . 278

gsl_siman_copy_t . . . . . . . . . . . . . . . . . . . . . . . . . . 278 gsl_siman_destroy_t. . . . . . . . . . . . . . . . . . . . . . . 278 gsl_siman_Efunc_t . . . . . . . . . . . . . . . . . . . . . . . . . 278 gsl_siman_metric_t. . . . . . . . . . . . . . . . . . . . . . . . 278 gsl_siman_params_t. . . . . . . . . . . . . . . . . . . . . . . . 279 gsl_siman_print_t . . . . . . . . . . . . . . . . . . . . . . . . . 278 gsl_siman_step_t . . . . . . . . . . . . . . . . . . . . . . . . . . 278

Appendix D: Concept Index 452

Concept Index

$

$, shell prompt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2

2D histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 2D random direction vector. . . . . . . . . . . . . . . . . . 211

3

3-j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3D random direction vector. . . . . . . . . . . . . . . . . . 211

6

6-j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9

9-j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A

acceleration of series . . . . . . . . . . . . . . . . . . . . . . . . 308 acosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Adaptive step-size control, differential equations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Ai(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Akima splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 aliasing of arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 alternative optimized functions . . . . . . . . . . . . . . . . 7 AMAX, Level-1 BLAS . . . . . . . . . . . . . . . . . . . . . . 112 angular reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ANSI C, use of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Apell symbol, see Pochammer symbol . . . . . . . . . 55 approximate comparison of floating point numbers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 arctangent integral. . . . . . . . . . . . . . . . . . . . . . . . . . . 52 argument of complex number . . . . . . . . . . . . . . . . . 21 arithmetic exceptions . . . . . . . . . . . . . . . . . . . . . . . 399 asinh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 astronomical constants . . . . . . . . . . . . . . . . . . . . . . 389 ASUM, Level-1 BLAS. . . . . . . . . . . . . . . . . . . . . . . 112 atanh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 atomic physics, constants. . . . . . . . . . . . . . . . . . . . 389 autoconf, using with GSL . . . . . . . . . . . . . . . . . . . 409 AXPY, Level-1 BLAS . . . . . . . . . . . . . . . . . . . . . . . 113

B

B-spline wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Bader and Deuflhard, Bulirsch-Stoer method.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Basic Linear Algebra Subroutines (BLAS) … 110, Bernoulli trial, random variates . . . . . . . . . . . . . . 219 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Bessel Functions, Fractional Order . . . . . . . . . . . . 39 best-fit parameters, covariance . . . . . . . . . . . . . . . 381 Beta distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Beta function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Beta function, incomplete normalized . . . . . . . . . 56 BFGS conjugate gradient algorithm, minimization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Bi(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 bias, IEEE format . . . . . . . . . . . . . . . . . . . . . . . . . . 397 bidiagonalization of real matrices . . . . . . . . . . . . 130 binning data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Binomial random variates . . . . . . . . . . . . . . . . . . . 220 biorthogonal wavelets . . . . . . . . . . . . . . . . . . . . . . . 312 bisection algorithm for finding roots . . . . . . . . . . 327 Bivariate Gaussian distribution . . . . . . . . . . . . . . 192 BLAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 BLAS, Low-level C interface . . . . . . . . . . . . . . . . . 411 blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 bounds checking, extension to GCC . . . . . . . . . . . 73 breakpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Brent’s method for finding minima . . . . . . . . . . . 338 Brent’s method for finding roots . . . . . . . . . . . . . 327 Broyden algorithm for multidimensional roots

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 BSD random number generator . . . . . . . . . . . . . . 175 bug-gsl mailing list. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 bugs, how to report . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Bulirsch-Stoer method . . . . . . . . . . . . . . . . . . . . . . 287

C

C extensions, compatible use of . . . . . . . . . . . . . . . . 4 C++, compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Carlson forms of Elliptic integrals . . . . . . . . . . . . . 46 Cash-Karp, Runge-Kutta method . . . . . . . . . . . . 287 Cauchy distribution . . . . . . . . . . . . . . . . . . . . . . . . . 196 Cauchy principal value, by numerical quadrature

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 CBLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 CBLAS, Low-level interface . . . . . . . . . . . . . . . . . 411 CDFs, cumulative distribution functions . . . . . . 187 Chebyshev series. . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 checking combination for validity . . . . . . . . . . . . 101 checking permutation for validity . . . . . . . . . . . . . 94 Chi(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chi-squared distribution . . . . . . . . . . . . . . . . . . . . 205 Cholesky decomposition . . . . . . . . . . . . . . . . . . . . . 128 Ci(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Clausen functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Clenshaw-Curtis quadrature . . . . . . . . . . . . . . . . . 158 CMRG, combined multiple recursive random

number generator . . . . . . . . . . . . . . . . . . . . . . 173 code reuse in applications . . . . . . . . . . . . . . . . . . . . 10 combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 combinatorial factor C(m,n) . . . . . . . . . . . . . . . . . . 55 combinatorial optimization . . . . . . . . . . . . . . . . . . 277 comparison functions, definition. . . . . . . . . . . . . . 104 compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 compiling programs, include paths . . . . . . . . . . . . . 4 compiling programs, library paths . . . . . . . . . . . . . . 4 complementary incomplete Gamma function … 56 complete Fermi-Dirac integrals. . . . . . . . . . . . . . . . 52 complex arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 21 complex cosine function, special functions . . . . . 66 Complex Gamma function . . . . . . . . . . . . . . . . . . . . 54 complex hermitian matrix, eigensystem . . . . . . . 136 complex log sine function, special functions …. 66 complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 complex sinc function, special functions . . . . . . . 66 complex sine function, special functions . . . . . . . 66 confluent hypergeometric function. . . . . . . . . . . . . 59 confluent hypergeometric functions . . . . . . . . . . . . 57 conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Conjugate gradient algorithm, minimization . . 360 conjugate of complex number . . . . . . . . . . . . . . . . . 22 constant matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 constants, fundamental. . . . . . . . . . . . . . . . . . . . . . 388 constants, mathematical—defined as macros … 16 constants, physical . . . . . . . . . . . . . . . . . . . . . . . . . . 388 constants, prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . 394 contacting the GSL developers . . . . . . . . . . . . . . . . . 3 conventions, used in manual . . . . . . . . . . . . . . . . . . . 3 convergence, accelerating a series . . . . . . . . . . . . 308 conversion of units . . . . . . . . . . . . . . . . . . . . . . . . . . 388 cooling schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 COPY, Level-1 BLAS . . . . . . . . . . . . . . . . . . . . . . . 112 cosine function, special functions . . . . . . . . . . . . . . 65 cosine of complex number . . . . . . . . . . . . . . . . . . . . 23 cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Coulomb wave functions. . . . . . . . . . . . . . . . . . . . . . 41 coupling coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 43 covariance matrix, from linear regression . . . . . 367 covariance matrix, linear fits . . . . . . . . . . . . . . . . . 366 covariance matrix, nonlinear fits . . . . . . . . . . . . . 381 covariance, of two datasets . . . . . . . . . . . . . . . . . . 235 CRAY random number generator, RANF . . . . . 177 cubic equation, solving . . . . . . . . . . . . . . . . . . . . . . . 28 cubic splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 cumulative distribution functions (CDFs) . . . . . 187 Cylindrical Bessel Functions . . . . . . . . . . . . . . . . . . 34

D

Daubechies wavelets . . . . . . . . . . . . . . . . . . . . . . . . 312 Dawson function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 debugging numerical programs . . . . . . . . . . . . . . . 403 Debye functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 denormalized form, IEEE format. . . . . . . . . . . . . 397 deprecated functions . . . . . . . . . . . . . . . . . . . . . . . . . . 9 derivatives, calculating numerically. . . . . . . . . . . 301 determinant of a matrix, by LU decomposition

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Deuflhard and Bader, Bulirsch-Stoer method.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 DFTs, see FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 diagonal, of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . 86 differential equations, initial value problems… 285 differentiation of functions, numeric . . . . . . . . . . 301 digamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 dilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 direction vector, random 2D . . . . . . . . . . . . . . . . . 211 direction vector, random 3D . . . . . . . . . . . . . . . . . 211 direction vector, random N-dimensional . . . . . . 211 Dirichlet distribution. . . . . . . . . . . . . . . . . . . . . . . . 215 Discrete Fourier Transforms, see FFT . . . . . . . . 140 discrete Hankel transforms . . . . . . . . . . . . . . . . . . 319 Discrete Newton algorithm for multidimensional

roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Discrete random numbers . . . . . . . . . . . . . . . 216, 217 Discrete random numbers, preprocessing. . . . . . 216 divided differences, polynomials . . . . . . . . . . . . . . . 27 division by zero, IEEE exceptions . . . . . . . . . . . . 399 dollar sign $, shell prompt . . . . . . . . . . . . . . . . . . . . . 3 DOT, Level-1 BLAS . . . . . . . . . . . . . . . . . . . . . . . . 111 double factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 double precision, IEEE format . . . . . . . . . . . . . . . 397 downloading GSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 DWT initialization. . . . . . . . . . . . . . . . . . . . . . . . . . 312 DWT, mathematical definition. . . . . . . . . . . . . . . 312 DWT, one dimensional . . . . . . . . . . . . . . . . . . . . . . 313 DWT, see wavelet transforms . . . . . . . . . . . . . . . . 312 DWT, two dimensional. . . . . . . . . . . . . . . . . . . . . . 314

E

e, defined as a macro. . . . . . . . . . . . . . . . . . . . . . . . . 16 E1(x), E2(x), Ei(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 51 eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . 135 elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . 16 elementary operations . . . . . . . . . . . . . . . . . . . . . . . . 45 elliptic functions (Jacobi). . . . . . . . . . . . . . . . . . . . . 48 elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 energy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 energy, units of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 erf(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 erfc(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Erlang distribution . . . . . . . . . . . . . . . . . . . . . . . . . 202 error codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Error handlers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 error handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 error handling macros . . . . . . . . . . . . . . . . . . . . . . . . 13 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 estimated standard deviation . . . . . . . . . . . . . . . . 232 estimated variance . . . . . . . . . . . . . . . . . . . . . . . . . . 232 euclidean distance function, hypot . . . . . . . . . . . . 17 Euler’s constant, defined as a macro . . . . . . . . . . . 16 evaluation of polynomials. . . . . . . . . . . . . . . . . . . . . 27 evaluation of polynomials, in divided difference

form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 examples, conventions used in. . . . . . . . . . . . . . . . . . 3 exceptions, IEEE arithmetic . . . . . . . . . . . . . . . . . 399 exchanging permutation elements . . . . . . . . . . . . . 94 exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 expm1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 exponent, IEEE format . . . . . . . . . . . . . . . . . . . . . 397 Exponential distribution . . . . . . . . . . . . . . . . . . . . 193 exponential function . . . . . . . . . . . . . . . . . . . . . . . . . 49 exponential integrals . . . . . . . . . . . . . . . . . . . . . . . . . 51 Exponential power distribution . . . . . . . . . . . . . . 195 exponential, difference from 1 computed accurately

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 exponentiation of complex number . . . . . . . . . . . . 22

F

F-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 factorial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 factorization of matrices. . . . . . . . . . . . . . . . . . . . . 122 false position algorithm for finding roots . . . . . . 327 Fast Fourier Transforms, see FFT . . . . . . . . . . . . 140 FDL, GNU Free Documentation License . . . . . . 430 Fehlberg method, differential equations . . . . . . . 287 Fermi-Dirac function . . . . . . . . . . . . . . . . . . . . . . . . . 52 FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 FFT mathematical definition . . . . . . . . . . . . . . . . 140 FFT of complex data, mixed-radix algorithm. . 144 FFT of complex data, radix-2 algorithm . . . . . . 142 FFT of real data . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 FFT of real data, mixed-radix algorithm. . . . . . 150 FFT of real data, radix-2 algorithm . . . . . . . . . . 149 FFT, complex data . . . . . . . . . . . . . . . . . . . . . . . . . 141 finding minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 finding roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 finding zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 fits, multi-parameter linear . . . . . . . . . . . . . . . . . . 368 fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 fitting, using Chebyshev polynomials . . . . . . . . . 304 Fj(x), Fermi-Dirac integral . . . . . . . . . . . . . . . . . . . 52 Fj(x,b), incomplete Fermi-Dirac integral . . . . . . . 53 flat distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Fletcher-Reeves conjugate gradient algorithm,

minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 floating point numbers, approximate comparison

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 force and energy, units of . . . . . . . . . . . . . . . . . . . . 394 Fortran range checking, equivalent in gcc . . . . . . 73 Four-tap Generalized Feedback Shift Register

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Fourier integrals, numerical . . . . . . . . . . . . . . . . . . 164 Fourier Transforms, see FFT . . . . . . . . . . . . . . . . 140 Fractional Order Bessel Functions . . . . . . . . . . . . . 39 free documentation . . . . . . . . . . . . . . . . . . . . . . . . . 424 free software, explanation of . . . . . . . . . . . . . . . . . . . 1 frexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 functions, numerical differentiation . . . . . . . . . . . 301 fundamental constants . . . . . . . . . . . . . . . . . . . . . . 388

G

Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . 202 gamma functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Gauss-Kronrod quadrature . . . . . . . . . . . . . . . . . . 158 Gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . 189 Gaussian distribution, bivariate . . . . . . . . . . . . . . 192 Gaussian Tail distribution . . . . . . . . . . . . . . . . . . . 191 gcc extensions, range-checking . . . . . . . . . . . . . . . . 73 gcc warning options . . . . . . . . . . . . . . . . . . . . . . . . . 405 gdb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Gear method, differential equations . . . . . . . . . . 287 Gegenbauer functions . . . . . . . . . . . . . . . . . . . . . . . . 57 GEMM, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . 117 GEMV, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . 114 general polynomial equations, solving . . . . . . . . . 28 Geometric random variates. . . . . . . . . . . . . . 224, 225 GER, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . . . 115 GERC, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . . 115 GERU, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . . 115 Givens Rotation, BLAS . . . . . . . . . . . . . . . . . . . . . 113 Givens Rotation, Modified, BLAS . . . . . . . . . . . . 113 GNU General Public License . . . . . . . . . . . . . . . . . . 1 golden section algorithm for finding minima… 338 gsl sf result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 gsl sf result e10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Gumbel distribution (Type 1) . . . . . . . . . . . . . . . 213 Gumbel distribution (Type 2) . . . . . . . . . . . . . . . 214

H

Haar wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Hankel transforms, discrete . . . . . . . . . . . . . . . . . . 319 HAVE INLINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 hazard function, normal distribution . . . . . . . . . . 49 HBOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 header files, including . . . . . . . . . . . . . . . . . . . . . . . . . 4 heapsort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 HEMM, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . 117 HEMV, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . 115 HER, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . . . 116 HER2, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . . 116 HER2K, Level-3 BLAS. . . . . . . . . . . . . . . . . . . . . . 120 HERK, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . . 119 hermitian matrix, complex, eigensystem . . . . . . 136 histogram statistics . . . . . . . . . . . . . . . . . . . . . . . . . 245 histogram, from ntuple . . . . . . . . . . . . . . . . . . . . . . 261 histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 histograms, random sampling from . . . . . . . . . . . 247 Householder linear solver . . . . . . . . . . . . . . . . . . . . 131 Householder matrix . . . . . . . . . . . . . . . . . . . . . . . . . 130 Householder transformation . . . . . . . . . . . . . . . . . 130 HYBRID algorithm, unscaled without derivatives

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 HYBRID algorithms for nonlinear systems. … 347 HYBRIDJ algorithm . . . . . . . . . . . . . . . . . . . . . . . . 347 HYBRIDS algorithm, scaled without derivatives

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 HYBRIDSJ algorithm. . . . . . . . . . . . . . . . . . . . . . . 347 hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 hyperbolic cosine, inverse . . . . . . . . . . . . . . . . . . . . . 17 hyperbolic functions, complex numbers . . . . . . . . 24 hyperbolic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 51 hyperbolic sine, inverse . . . . . . . . . . . . . . . . . . . . . . . 17 hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 hyperbolic tangent, inverse . . . . . . . . . . . . . . . . . . . 17 hypergeometric functions . . . . . . . . . . . . . . . . . . . . . 57 hypergeometric random variates . . . . . . . . . . . . . 225 hypot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 hypot function, special functions . . . . . . . . . . . . . . 65

I

identity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 identity permutation . . . . . . . . . . . . . . . . . . . . . . . . . 93 IEEE exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 IEEE floating point . . . . . . . . . . . . . . . . . . . . . . . . . 397 IEEE format for floating point numbers . . . . . . 397 IEEE infinity, defined as a macro . . . . . . . . . . . . . 16 IEEE NaN, defined as a macro. . . . . . . . . . . . . . . . 16 illumination, units of. . . . . . . . . . . . . . . . . . . . . . . . 393 imperial units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 importance sampling, VEGAS . . . . . . . . . . . . . . . 270 including GSL header files . . . . . . . . . . . . . . . . . . . . . 4 incomplete Beta function, normalized . . . . . . . . . 56 incomplete Fermi-Dirac integral . . . . . . . . . . . . . . . 53 incomplete Gamma function . . . . . . . . . . . . . . . . . . 56 indirect sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 indirect sorting, of vector elements . . . . . . . . . . . 105 infinity, defined as a macro . . . . . . . . . . . . . . . . . . . 16 infinity, IEEE format . . . . . . . . . . . . . . . . . . . . . . . 397 info-gsl mailing list. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 initial value problems, differential equations… 285 initializing matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 82 initializing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 inline functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 integer powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 integrals, exponential . . . . . . . . . . . . . . . . . . . . . . . . 51 integration, numerical (quadrature) . . . . . . . . . . 157 interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 interpolation, using Chebyshev polynomials … 304 inverse complex trigonometric functions . . . . . . . 23 inverse cumulative distribution functions . . . . . 187 inverse hyperbolic cosine . . . . . . . . . . . . . . . . . . . . . 17 inverse hyperbolic functions, complex numbers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 inverse hyperbolic sine . . . . . . . . . . . . . . . . . . . . . . . 17 inverse hyperbolic tangent . . . . . . . . . . . . . . . . . . . . 17 inverse of a matrix, by LU decomposition . . . . . 123 inverting a permutation . . . . . . . . . . . . . . . . . . . . . . 94 Irregular Cylindrical Bessel Functions . . . . . . . . . 34 Irregular Modified Bessel Functions, Fractional

Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Irregular Modified Cylindrical Bessel Functions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Irregular Modified Spherical Bessel Functions . . 39 Irregular Spherical Bessel Functions . . . . . . . . . . . 38 iterating through combinations . . . . . . . . . . . . . . 101 iterating through permutations . . . . . . . . . . . . . . . 94 iterative refinement of solutions in linear systems

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

J

Jacobi elliptic functions . . . . . . . . . . . . . . . . . . . . . . 48

L

Laguerre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Lambert function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Landau distribution. . . . . . . . . . . . . . . . . . . . . . . . . 199 LAPACK, recommended for linear algebra … 122,

135 Laplace distribution. . . . . . . . . . . . . . . . . . . . . . . . . 194 LD LIBRARY PATH . . . . . . . . . . . . . . . . . . . . . . . . . 5 ldexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 leading dimension, matrices. . . . . . . . . . . . . . . . . . . 80 least squares fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 least squares fitting, nonlinear . . . . . . . . . . . . . . . 376 least squares, covariance of best-fit parameters

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Legendre forms of elliptic integrals . . . . . . . . . . . . 46 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 length, computed accurately using hypot . . . . . . 17 Levenberg-Marquardt algorithms. . . . . . . . . . . . . 380 Levin u-transform . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Levy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Levy distribution, skew . . . . . . . . . . . . . . . . . . . . . 201 libraries, linking with . . . . . . . . . . . . . . . . . . . . . . . . . 4 libraries, shared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 license of GSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 light, units of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 linear algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 linear algebra, BLAS. . . . . . . . . . . . . . . . . . . . . . . . 110 linear interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 294 linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 linear systems, refinement of solutions . . . . . . . . 123 linear systems, solution of . . . . . . . . . . . . . . . . . . . 122 linking with GSL libraries . . . . . . . . . . . . . . . . . . . . . 4 LMDER algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 380 log1p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 logarithm and related functions . . . . . . . . . . . . . . . 63 logarithm of Beta function. . . . . . . . . . . . . . . . . . . . 56 logarithm of combinatorial factor C(m,n) . . . . . . 55 logarithm of complex number . . . . . . . . . . . . . . . . . 23 logarithm of cosh function, special functions … 66 logarithm of double factorial . . . . . . . . . . . . . . . . . . 55 logarithm of factorial. . . . . . . . . . . . . . . . . . . . . . . . . 55 logarithm of Gamma function. . . . . . . . . . . . . . . . . 54 logarithm of Pochhammer symbol . . . . . . . . . . . . . 55 logarithm of sinh function, special functions … 66 logarithm of the determinant of a matrix . . . . . 123 logarithm, computed accurately near 1 . . . . . . . . 17 Logarithmic random variates . . . . . . . . . . . . . . . . 226 Logistic distribution . . . . . . . . . . . . . . . . . . . . . . . . 209 Lognormal distribution . . . . . . . . . . . . . . . . . . . . . . 204 long double. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 low discrepancy sequences . . . . . . . . . . . . . . . . . . . 184 Low-level CBLAS. . . . . . . . . . . . . . . . . . . . . . . . . . . 411 LU decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 122

M

macros for mathematical constants . . . . . . . . . . . . 16 magnitude of complex number . . . . . . . . . . . . . . . . 21 mailing list archives . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 mailing list for GSL announcements . . . . . . . . . . . . 2 mailing list, bug-gsl . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 mantissa, IEEE format . . . . . . . . . . . . . . . . . . . . . . 397 mass, units of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 mathematical constants, defined as macros. . . . . 16 mathematical functions, elementary . . . . . . . . . . . 16 matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 80 matrices, initializing . . . . . . . . . . . . . . . . . . . . . . . . . 82 matrices, range-checking. . . . . . . . . . . . . . . . . . . . . . 82 matrix determinant . . . . . . . . . . . . . . . . . . . . . . . . . 123 matrix diagonal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 matrix factorization. . . . . . . . . . . . . . . . . . . . . . . . . 122 matrix inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 matrix square root, Cholesky decomposition . . 128 matrix subdiagonal . . . . . . . . . . . . . . . . . . . . . . . . . . 86 matrix superdiagonal. . . . . . . . . . . . . . . . . . . . . . . . . 86 matrix, constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 matrix, identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 matrix, operations . . . . . . . . . . . . . . . . . . . . . . . . . . 110 matrix, zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 maximal phase, Daubechies wavelets . . . . . . . . . 312 maximization, see minimization . . . . . . . . . . . . . . 334 maximum of two numbers . . . . . . . . . . . . . . . . . . . . 19 maximum value, from histogram . . . . . . . . . . . . . 245 mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 mean value, from histogram . . . . . . . . . . . . . . . . . 245 Mill’s ratio, inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 49 min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 minimization, BFGS conjugate gradient algorithm

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 minimization, caveats . . . . . . . . . . . . . . . . . . . . . . . 335 minimization, conjugate gradient algorithm … 360 minimization, multidimensional . . . . . . . . . . . . . . 355 minimization, one-dimensional . . . . . . . . . . . . . . . 334 minimization, overview . . . . . . . . . . . . . . . . . . . . . . 334 minimization, Polak-Ribiere algorithm. . . . . . . . 360 minimization, providing a function to minimize

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 minimization, simplex algorithm . . . . . . . . . . . . . 361 minimization, steepest descent algorithm . . . . . 360 minimization, stopping parameters . . . . . . . . . . . 337 minimum finding, Brent’s method . . . . . . . . . . . . 338 minimum finding, golden section algorithm … 338 minimum of two numbers . . . . . . . . . . . . . . . . . . . . 19 minimum value, from histogram . . . . . . . . . . . . . 245 MINPACK, minimization algorithms. . . . . 347, 380 MISCFUN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 MISER monte carlo integration . . . . . . . . . . . . . . 268 Mixed-radix FFT, complex data . . . . . . . . . . . . . 144 Mixed-radix FFT, real data . . . . . . . . . . . . . . . . . 150 Modified Bessel Functions, Fractional Order … 40 Modified Clenshaw-Curtis quadrature . . . . . . . . 158 Modified Cylindrical Bessel Functions . . . . . . . . . 35 Modified Givens Rotation, BLAS . . . . . . . . . . . . 113 Modified Newton’s method for nonlinear systems

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Modified Spherical Bessel Functions . . . . . . . . . . . 38 Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . 266 MRG, multiple recursive random number generator

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 MT19937 random number generator. . . . . . . . . . 172 multi-parameter regression . . . . . . . . . . . . . . . . . . 368 multidimensional root finding, Broyden algorithm

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 multidimensional root finding, overview . . . . . . 341 multidimensional root finding, providing a function to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Multimin, caveats. . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Multinomial distribution . . . . . . . . . . . . . . . . . . . . 221 multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

N

N-dimensional random direction vector . . . . . . . 211 NaN, defined as a macro . . . . . . . . . . . . . . . . . . . . . 16 nautical units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Negative Binomial distribution, random variates

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Nelder-Mead simplex algorithm for minimization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Newton algorithm, discrete . . . . . . . . . . . . . . . . . . 348 Newton algorithm, globally convergent . . . . . . . 348 Newton’s method for finding roots . . . . . . . . . . . 328 Newton’s method for systems of nonlinear

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Niederreiter sequence . . . . . . . . . . . . . . . . . . . . . . . 184 NIST Statistical Reference Datasets . . . . . . . . . . 375 non-normalized incomplete Gamma function … 56 nonlinear equation, solutions of . . . . . . . . . . . . . . 321 nonlinear fitting, stopping parameters . . . . . . . . 379 nonlinear functions, minimization . . . . . . . . . . . . 334 nonlinear least squares fitting . . . . . . . . . . . . . . . . 376 nonlinear least squares fitting, overview . . . . . . 376 nonlinear systems of equations, solution of …. 341 normalized form, IEEE format . . . . . . . . . . . . . . . 397 normalized incomplete Beta function . . . . . . . . . . 56 Not-a-number, defined as a macro. . . . . . . . . . . . . 16 NRM2, Level-1 BLAS . . . . . . . . . . . . . . . . . . . . . . . 112 ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 nuclear physics, constants . . . . . . . . . . . . . . . . . . . 389 numerical constants, defined as macros . . . . . . . . 16 numerical derivatives. . . . . . . . . . . . . . . . . . . . . . . . 301 numerical integration (quadrature) . . . . . . . . . . . 157

O

obtaining GSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ODEs, initial value problems . . . . . . . . . . . . . . . . 285 optimization, combinatorial. . . . . . . . . . . . . . . . . . 277 optimization, see minimization . . . . . . . . . . . . . . . 334 optimized functions, alternatives . . . . . . . . . . . . . . . 7 ordinary differential equations, initial value

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 oscillatory functions, numerical integration of

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 overflow, IEEE exceptions . . . . . . . . . . . . . . . . . . . 399

P

Pareto distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 210 PAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . 388 physical dimension, matrices . . . . . . . . . . . . . . . . . . 80 pi, defined as a macro . . . . . . . . . . . . . . . . . . . . . . . . 16 plain monte carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Pochhammer symbol . . . . . . . . . . . . . . . . . . . . . . . . . 55 Poisson random numbers . . . . . . . . . . . . . . . . . . . . 218 Polak-Ribiere algorithm, minimization. . . . . . . . 360 polar form of complex numbers . . . . . . . . . . . . . . . 20 polar to rectangular conversion . . . . . . . . . . . . . . . 66 polygamma functions . . . . . . . . . . . . . . . . . . . . . . . . 64 polynomial evaluation . . . . . . . . . . . . . . . . . . . . . . . . 27 polynomial interpolation . . . . . . . . . . . . . . . . . . . . 294 polynomials, roots of . . . . . . . . . . . . . . . . . . . . . . . . . 27 power function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 power of complex number . . . . . . . . . . . . . . . . . . . . 22 power, units of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 precision, IEEE arithmetic . . . . . . . . . . . . . . . . . . 399 prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 pressure, units of . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Prince-Dormand, Runge-Kutta method . . . . . . . 287 printers units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 probability distribution, from histogram . . . . . . 247 probability distributions, from histograms …. 247 projection of ntuples . . . . . . . . . . . . . . . . . . . . . . . . 261 psi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Q

QR decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 124 QR decomposition with column pivoting. . . . . . 125 QUADPACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 quadratic equation, solving . . . . . . . . . . . . . . . . . . . 27 quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 quantile functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 187 quasi-random sequences . . . . . . . . . . . . . . . . . . . . . 184

R

R250 shift-register random number generator. . 177 Racah coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 radioactivity, units of . . . . . . . . . . . . . . . . . . . . . . . 393 Radix-2 FFT for real data . . . . . . . . . . . . . . . . . . . 149 Radix-2 FFT, complex data . . . . . . . . . . . . . . . . . 142 rand, BSD random number generator . . . . . . . . 175 rand48 random number generator . . . . . . . . . . . . 176 random number distributions . . . . . . . . . . . . . . . . 187 random number generators . . . . . . . . . . . . . . . . . . 167 random sampling from histograms . . . . . . . . . . . 247 RANDU random number generator . . . . . . . . . . 178 RANF random number generator . . . . . . . . . . . . 177 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 range-checking for matrices . . . . . . . . . . . . . . . . . . . 82 range-checking for vectors . . . . . . . . . . . . . . . . . . . . 73 RANLUX random number generator . . . . . . . . . 173 RANLXD random number generator . . . . . . . . . 173 RANLXS random number generator . . . . . . . . . 173 RANMAR random number generator …. 177, 178 Rayleigh distribution. . . . . . . . . . . . . . . . . . . . . . . . 197 Rayleigh Tail distribution . . . . . . . . . . . . . . . . . . . 198 real symmetric matrix, eigensystem . . . . . . . . . . 135 Reciprocal Gamma function . . . . . . . . . . . . . . . . . . 54 rectangular to polar conversion . . . . . . . . . . . . . . . 66 recursive stratified sampling, MISER . . . . . . . . . 268 reduction of angular variables. . . . . . . . . . . . . . . . . 66 refinement of solutions in linear systems . . . . . . 123 regression, least squares . . . . . . . . . . . . . . . . . . . . . 366 Regular Bessel Functions, Fractional Order …. 39 Regular Bessel Functions, Zeros of . . . . . . . . . . . . 41 Regular Cylindrical Bessel Functions . . . . . . . . . . 34 Regular Modified Bessel Functions, Fractional

Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Regular Modified Cylindrical Bessel Functions. . 35 Regular Modified Spherical Bessel Functions … 38 Regular Spherical Bessel Functions . . . . . . . . . . . . 37 Regulated Gamma function. . . . . . . . . . . . . . . . . . . 54 relative Pochhammer symbol . . . . . . . . . . . . . . . . . 56 reporting bugs in GSL . . . . . . . . . . . . . . . . . . . . . . . . 3 representations of complex numbers . . . . . . . . . . . 20 resampling from histograms . . . . . . . . . . . . . . . . . 247 residual, in nonlinear systems of equations … 346,

379 reversing a permutation . . . . . . . . . . . . . . . . . . . . . . 94 RK2, Runge-Kutta method . . . . . . . . . . . . . . . . . . 286 RK4, Runge-Kutta method . . . . . . . . . . . . . . . . . . 287 RKF45, Runge-Kutta-Fehlberg method. . . . . . . 287 root finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 root finding, bisection algorithm . . . . . . . . . . . . . 327 root finding, Brent’s method. . . . . . . . . . . . . . . . . 327 root finding, caveats . . . . . . . . . . . . . . . . . . . . . . . . 321 root finding, false position algorithm . . . . . . . . . 327 root finding, initial guess . . . . . . . . . . . . . . . . . . . . 325 root finding, Newton’s method . . . . . . . . . . . . . . . 328 root finding, overview . . . . . . . . . . . . . . . . . . . . . . . 321 root finding, providing a function to solve …. 323 root finding, search bounds . . . . . . . . . . . . . . . . . . 325 root finding, secant method. . . . . . . . . . . . . . . . . . 328 root finding, Steffenson’s method . . . . . . . . . . . . 329 root finding, stopping parameters. . . . . . . . 326, 346 roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 ROTG, Level-1 BLAS. . . . . . . . . . . . . . . . . . . . . . . 113 rounding mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Runge-Kutta Cash-Karp method. . . . . . . . . . . . . 287 Runge-Kutta methods, ordinary differential

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Runge-Kutta Prince-Dormand method . . . . . . . 287

S

safe comparison of floating point numbers . . . . . 19 sampling from histograms . . . . . . . . . . . . . . . . . . . 247 SCAL, Level-1 BLAS . . . . . . . . . . . . . . . . . . . . . . . 113 schedule, cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 secant method for finding roots . . . . . . . . . . . . . . 328 selection function, ntuples . . . . . . . . . . . . . . . . . . . 261 series, acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 308 shared libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 shell prompt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Shi(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 shift-register random number generator. . . . . . . 177 Si(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 sign bit, IEEE format . . . . . . . . . . . . . . . . . . . . . . . 397 sign of the determinant of a matrix . . . . . . . . . . 123 simplex algorithm, minimization . . . . . . . . . . . . . 361 simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . 277 sin, of complex number. . . . . . . . . . . . . . . . . . . . . . . 23 sine function, special functions . . . . . . . . . . . . . . . . 65 single precision, IEEE format . . . . . . . . . . . . . . . . 397 singular functions, numerical integration of … 161 singular points, specifying positions in quadrature

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 singular value decomposition . . . . . . . . . . . . . . . . 127 Skew Levy distribution . . . . . . . . . . . . . . . . . . . . . . 201 slope, see numerical derivative . . . . . . . . . . . . . . . 301 Sobol sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 solution of linear system by Householder

transformations . . . . . . . . . . . . . . . . . . . . . . . . 131 solution of linear systems, Ax=b . . . . . . . . . . . . . 122 solving a nonlinear equation . . . . . . . . . . . . . . . . . 321 solving nonlinear systems of equations . . . . . . . . 341 sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 sorting eigenvalues and eigenvectors . . . . . . . . . . 136 sorting vector elements . . . . . . . . . . . . . . . . . . . . . . 105 source code, reuse in applications . . . . . . . . . . . . . 10 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . 37 spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . 60 spherical random variates, 2D . . . . . . . . . . . . . . . 211 spherical random variates, 3D . . . . . . . . . . . . . . . 211 spherical random variates, N-dimensional . . . . . 211 spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 square root of a matrix, Cholesky decomposition

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 square root of complex number . . . . . . . . . . . . . . . 22 standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . 232 standard deviation, from histogram . . . . . . . . . . 245 standards conformance, ANSI C . . . . . . . . . . . . . . . 4 Statistical Reference Datasets (StRD) . . . . . . . . 375 statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 statistics, from histogram. . . . . . . . . . . . . . . . . . . . 245 steepest descent algorithm, minimization . . . . . 360 Steffenson’s method for finding roots . . . . . . . . . 329 stratified sampling in monte carlo integration. . 266 stride, of vector index . . . . . . . . . . . . . . . . . . . . . . . . 72 Student t-distribution . . . . . . . . . . . . . . . . . . . . . . . 207 subdiagonal, of a matrix. . . . . . . . . . . . . . . . . . . . . . 86 summation, acceleration. . . . . . . . . . . . . . . . . . . . . 308 superdiagonal, matrix . . . . . . . . . . . . . . . . . . . . . . . . 86 SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 SWAP, Level-1 BLAS . . . . . . . . . . . . . . . . . . . . . . . 112 swapping permutation elements . . . . . . . . . . . . . . . 94 SYMM, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . 117 symmetric matrix, real, eigensystem. . . . . . . . . . 135 SYMV, Level-2 BLAS. . . . . . . . . . . . . . . . . . . . . . . 115 synchrotron functions . . . . . . . . . . . . . . . . . . . . . . . . 64 SYR, Level-2 BLAS. . . . . . . . . . . . . . . . . . . . . . . . . 116 SYR2, Level-2 BLAS. . . . . . . . . . . . . . . . . . . . . . . . 116 SYR2K, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . 119 SYRK, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . . 119 systems of equations, nonlinear . . . . . . . . . . . . . . 341

T

t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 tangent of complex number . . . . . . . . . . . . . . . . . . . 23 Tausworthe random number generator. . . . . . . . 174 Taylor coefficients, computation of . . . . . . . . . . . . 55 testing combination for validity . . . . . . . . . . . . . . 101 testing permutation for validity . . . . . . . . . . . . . . . 94 thermal energy, units of . . . . . . . . . . . . . . . . . . . . . 392 time units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 trailing dimension, matrices . . . . . . . . . . . . . . . . . . 80 transformation, Householder . . . . . . . . . . . . . . . . . 130 transforms, Hankel. . . . . . . . . . . . . . . . . . . . . . . . . . 319 transforms, wavelet . . . . . . . . . . . . . . . . . . . . . . . . . 312 transport functions . . . . . . . . . . . . . . . . . . . . . . . . . . 65 traveling salesman problem . . . . . . . . . . . . . . . . . . 281 tridiagonal decomposition . . . . . . . . . . . . . . . 128, 129 tridiagonal systems . . . . . . . . . . . . . . . . . . . . . . . . . 131 trigonometric functions. . . . . . . . . . . . . . . . . . . . . . . 65 trigonometric functions of complex numbers … 23 trigonometric integrals . . . . . . . . . . . . . . . . . . . . . . . 52 TRMM, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . 118 TRMV, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . 114 TRSM, Level-3 BLAS . . . . . . . . . . . . . . . . . . . . . . . 118 TRSV, Level-2 BLAS . . . . . . . . . . . . . . . . . . . . . . . 114 TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 TT800 random number generator . . . . . . . . . . . . 178 two dimensional Gaussian distribution. . . . . . . . 192 two dimensional histograms . . . . . . . . . . . . . . . . . 250 two-sided exponential distribution. . . . . . . . . . . . 194 Type 1 Gumbel distribution, random variates

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Type 2 Gumbel distribution . . . . . . . . . . . . . . . . . 214

U

u-transform for series . . . . . . . . . . . . . . . . . . . . . . . 308 underflow, IEEE exceptions . . . . . . . . . . . . . . . . . 399 uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . 203 units, conversion of . . . . . . . . . . . . . . . . . . . . . . . . . 388 units, imperial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Unix random number generators, rand . . . . . . . 175 Unix random number generators, rand48 . . . . . 175 unnormalized incomplete Gamma function . . . . . 56 unweighted linear fits . . . . . . . . . . . . . . . . . . . . . . . 366 usage, compiling application programs . . . . . . . . . . 4

V

value function, ntuples . . . . . . . . . . . . . . . . . . . . . . 261 Van der Pol oscillator, example . . . . . . . . . . . . . . 290 variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 variance, from histogram . . . . . . . . . . . . . . . . . . . . 245 variance-covariance matrix, linear fits . . . . . . . . 366 VAX random number generator . . . . . . . . . . . . . . 178 vector, operations. . . . . . . . . . . . . . . . . . . . . . . . . . . 110 vector, sorting elements of . . . . . . . . . . . . . . . . . . . 105 vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 72 vectors, initializing. . . . . . . . . . . . . . . . . . . . . . . . . . . 74 vectors, range-checking . . . . . . . . . . . . . . . . . . . . . . . 73 VEGAS monte carlo integration . . . . . . . . . . . . . 270 viscosity, units of . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 volume units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

W

W function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 warning options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 warranty (none) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 wavelet transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 312 website, developer information . . . . . . . . . . . . . . . . . 3 Weibull distribution. . . . . . . . . . . . . . . . . . . . . . . . . 212 weight, units of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 weighted linear fits. . . . . . . . . . . . . . . . . . . . . . . . . . 366 Wigner coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Z

zero finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 zero matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 zero, IEEE format . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Zeros of Regular Bessel Functions . . . . . . . . . . . . . 41 Zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67