GNU Scientific Library Reference Manual Third edition, for GSL Version 1.12 January 2009 Mark Galassi Jim Davies James Theiler Brian Gough Gerard Jungman Patrick Alken Michael Booth Fabrice Rossi

GNU Scientific Library Reference Manual - Third Edition (v1.12) Buy the book here >>> support free documentation

This file documents the GNU Scientific Library (GSL), a collection of numerical routines for scientific computing. It corresponds to release 1.12 of the library. Please report any errors in this manual to bug-gsl at gnu.org.

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

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

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

• 1 Introduction
  • 1.1 Routines available in GSL
  • 1.2 GSL is Free Software
  • 1.3 Obtaining GSL
  • 1.4 No Warranty
  • 1.5 Reporting Bugs
  • 1.6 Further Information
  • 1.7 Conventions used in this manual
• 2 Using the library
  • 2.1 An Example Program
  • 2.2 Compiling and Linking
    • 2.2.1 Linking programs with the library
    • 2.2.2 Linking with an alternative BLAS library
  • 2.3 Shared Libraries
  • 2.4 ANSI C Compliance
  • 2.5 Inline functions
  • 2.6 Long double
  • 2.7 Portability functions
  • 2.8 Alternative optimized functions
  • 2.9 Support for different numeric types
  • 2.10 Compatibility with C++
  • 2.11 Aliasing of arrays
  • 2.12 Thread-safety
  • 2.13 Deprecated Functions
  • 2.14 Code Reuse
• 3 Error Handling
  • 3.1 Error Reporting
  • 3.2 Error Codes
  • 3.3 Error Handlers
  • 3.4 Using GSL error reporting in your own functions
  • 3.5 Examples
• 4 Mathematical Functions
  • 4.1 Mathematical Constants
  • 4.2 Infinities and Not-a-number
  • 4.3 Elementary Functions
  • 4.4 Small integer powers
  • 4.5 Testing the Sign of Numbers
  • 4.6 Testing for Odd and Even Numbers
  • 4.7 Maximum and Minimum functions
  • 4.8 Approximate Comparison of Floating Point Numbers
• 5 Complex Numbers
  • 5.1 Representation of complex numbers
  • 5.2 Properties of complex numbers
  • 5.3 Complex arithmetic operators
  • 5.4 Elementary Complex Functions
  • 5.5 Complex Trigonometric Functions
  • 5.6 Inverse Complex Trigonometric Functions
  • 5.7 Complex Hyperbolic Functions
  • 5.8 Inverse Complex Hyperbolic Functions
  • 5.9 References and Further Reading
• 6 Polynomials
  • 6.1 Polynomial Evaluation
  • 6.2 Divided Difference Representation of Polynomials
  • 6.3 Quadratic Equations
  • 6.4 Cubic Equations
  • 6.5 General Polynomial Equations
  • 6.6 Examples
  • 6.7 References and Further Reading
• 7 Special Functions
  • 7.1 Usage
  • 7.2 The gsl_sf_result struct
  • 7.3 Modes
  • 7.4 Airy Functions and Derivatives
    • 7.4.1 Airy Functions
    • 7.4.2 Derivatives of Airy Functions
    • 7.4.3 Zeros of Airy Functions
    • 7.4.4 Zeros of Derivatives of Airy Functions
  • 7.5 Bessel Functions
    • 7.5.1 Regular Cylindrical Bessel Functions
    • 7.5.2 Irregular Cylindrical Bessel Functions
    • 7.5.3 Regular Modified Cylindrical Bessel Functions
    • 7.5.4 Irregular Modified Cylindrical Bessel Functions
    • 7.5.5 Regular Spherical Bessel Functions
    • 7.5.6 Irregular Spherical Bessel Functions
    • 7.5.7 Regular Modified Spherical Bessel Functions
    • 7.5.8 Irregular Modified Spherical Bessel Functions
    • 7.5.9 Regular Bessel Function--Fractional Order
    • 7.5.10 Irregular Bessel Functions--Fractional Order
    • 7.5.11 Regular Modified Bessel Functions--Fractional Order
    • 7.5.12 Irregular Modified Bessel Functions--Fractional Order
    • 7.5.13 Zeros of Regular Bessel Functions
  • 7.6 Clausen Functions
  • 7.7 Coulomb Functions
    • 7.7.1 Normalized Hydrogenic Bound States
    • 7.7.2 Coulomb Wave Functions
    • 7.7.3 Coulomb Wave Function Normalization Constant
  • 7.8 Coupling Coefficients
    • 7.8.1 3-j Symbols
    • 7.8.2 6-j Symbols
    • 7.8.3 9-j Symbols
  • 7.9 Dawson Function
  • 7.10 Debye Functions
  • 7.11 Dilogarithm
    • 7.11.1 Real Argument
    • 7.11.2 Complex Argument
  • 7.12 Elementary Operations
  • 7.13 Elliptic Integrals
    • 7.13.1 Definition of Legendre Forms
    • 7.13.2 Definition of Carlson Forms
    • 7.13.3 Legendre Form of Complete Elliptic Integrals
    • 7.13.4 Legendre Form of Incomplete Elliptic Integrals
    • 7.13.5 Carlson Forms
  • 7.14 Elliptic Functions (Jacobi)
  • 7.15 Error Functions
    • 7.15.1 Error Function
    • 7.15.2 Complementary Error Function
    • 7.15.3 Log Complementary Error Function
    • 7.15.4 Probability functions
  • 7.16 Exponential Functions
    • 7.16.1 Exponential Function
    • 7.16.2 Relative Exponential Functions
    • 7.16.3 Exponentiation With Error Estimate
  • 7.17 Exponential Integrals
    • 7.17.1 Exponential Integral
    • 7.17.2 Ei(x)
    • 7.17.3 Hyperbolic Integrals
    • 7.17.4 Ei_3(x)
    • 7.17.5 Trigonometric Integrals
    • 7.17.6 Arctangent Integral
  • 7.18 Fermi-Dirac Function
    • 7.18.1 Complete Fermi-Dirac Integrals
    • 7.18.2 Incomplete Fermi-Dirac Integrals
  • 7.19 Gamma and Beta Functions
    • 7.19.1 Gamma Functions
    • 7.19.2 Factorials
    • 7.19.3 Pochhammer Symbol
    • 7.19.4 Incomplete Gamma Functions
    • 7.19.5 Beta Functions
    • 7.19.6 Incomplete Beta Function
  • 7.20 Gegenbauer Functions
  • 7.21 Hypergeometric Functions
  • 7.22 Laguerre Functions
  • 7.23 Lambert W Functions
  • 7.24 Legendre Functions and Spherical Harmonics
    • 7.24.1 Legendre Polynomials
    • 7.24.2 Associated Legendre Polynomials and Spherical Harmonics
    • 7.24.3 Conical Functions
    • 7.24.4 Radial Functions for Hyperbolic Space
  • 7.25 Logarithm and Related Functions
  • 7.26 Mathieu Functions
    • 7.26.1 Mathieu Function Workspace
    • 7.26.2 Mathieu Function Characteristic Values
    • 7.26.3 Angular Mathieu Functions
    • 7.26.4 Radial Mathieu Functions
  • 7.27 Power Function
  • 7.28 Psi (Digamma) Function
    • 7.28.1 Digamma Function
    • 7.28.2 Trigamma Function
    • 7.28.3 Polygamma Function
  • 7.29 Synchrotron Functions
  • 7.30 Transport Functions
  • 7.31 Trigonometric Functions
    • 7.31.1 Circular Trigonometric Functions
    • 7.31.2 Trigonometric Functions for Complex Arguments
    • 7.31.3 Hyperbolic Trigonometric Functions
    • 7.31.4 Conversion Functions
    • 7.31.5 Restriction Functions
    • 7.31.6 Trigonometric Functions With Error Estimates
  • 7.32 Zeta Functions
    • 7.32.1 Riemann Zeta Function
    • 7.32.2 Riemann Zeta Function Minus One
    • 7.32.3 Hurwitz Zeta Function
    • 7.32.4 Eta Function
  • 7.33 Examples
  • 7.34 References and Further Reading
• 8 Vectors and Matrices
  • 8.1 Data types
  • 8.2 Blocks
    • 8.2.1 Block allocation
    • 8.2.2 Reading and writing blocks
    • 8.2.3 Example programs for blocks
  • 8.3 Vectors
    • 8.3.1 Vector allocation
    • 8.3.2 Accessing vector elements
    • 8.3.3 Initializing vector elements
    • 8.3.4 Reading and writing vectors
    • 8.3.5 Vector views
    • 8.3.6 Copying vectors
    • 8.3.7 Exchanging elements
    • 8.3.8 Vector operations
    • 8.3.9 Finding maximum and minimum elements of vectors
    • 8.3.10 Vector properties
    • 8.3.11 Example programs for vectors
  • 8.4 Matrices
    • 8.4.1 Matrix allocation
    • 8.4.2 Accessing matrix elements
    • 8.4.3 Initializing matrix elements
    • 8.4.4 Reading and writing matrices
    • 8.4.5 Matrix views
    • 8.4.6 Creating row and column views
    • 8.4.7 Copying matrices
    • 8.4.8 Copying rows and columns
    • 8.4.9 Exchanging rows and columns
    • 8.4.10 Matrix operations
    • 8.4.11 Finding maximum and minimum elements of matrices
    • 8.4.12 Matrix properties
    • 8.4.13 Example programs for matrices
  • 8.5 References and Further Reading
• 9 Permutations
  • 9.1 The Permutation struct
  • 9.2 Permutation allocation
  • 9.3 Accessing permutation elements
  • 9.4 Permutation properties
  • 9.5 Permutation functions
  • 9.6 Applying Permutations
  • 9.7 Reading and writing permutations
  • 9.8 Permutations in cyclic form
  • 9.9 Examples
  • 9.10 References and Further Reading
• 10 Combinations
  • 10.1 The Combination struct
  • 10.2 Combination allocation
  • 10.3 Accessing combination elements
  • 10.4 Combination properties
  • 10.5 Combination functions
  • 10.6 Reading and writing combinations
  • 10.7 Examples
  • 10.8 References and Further Reading
• 11 Sorting
  • 11.1 Sorting objects
  • 11.2 Sorting vectors
  • 11.3 Selecting the k smallest or largest elements
  • 11.4 Computing the rank
  • 11.5 Examples
  • 11.6 References and Further Reading
• 12 BLAS Support
  • 12.1 GSL BLAS Interface
    • 12.1.1 Level 1
    • 12.1.2 Level 2
    • 12.1.3 Level 3
  • 12.2 Examples
  • 12.3 References and Further Reading
• 13 Linear Algebra
  • 13.1 LU Decomposition
  • 13.2 QR Decomposition
  • 13.3 QR Decomposition with Column Pivoting
  • 13.4 Singular Value Decomposition
  • 13.5 Cholesky Decomposition
  • 13.6 Tridiagonal Decomposition of Real Symmetric Matrices
  • 13.7 Tridiagonal Decomposition of Hermitian Matrices
  • 13.8 Hessenberg Decomposition of Real Matrices
  • 13.9 Hessenberg-Triangular Decomposition of Real Matrices
  • 13.10 Bidiagonalization
  • 13.11 Householder Transformations
  • 13.12 Householder solver for linear systems
  • 13.13 Tridiagonal Systems
  • 13.14 Balancing
  • 13.15 Examples
  • 13.16 References and Further Reading
• 14 Eigensystems
  • 14.1 Real Symmetric Matrices
  • 14.2 Complex Hermitian Matrices
  • 14.3 Real Nonsymmetric Matrices
  • 14.4 Real Generalized Symmetric-Definite Eigensystems
  • 14.5 Complex Generalized Hermitian-Definite Eigensystems
  • 14.6 Real Generalized Nonsymmetric Eigensystems
  • 14.7 Sorting Eigenvalues and Eigenvectors
  • 14.8 Examples
  • 14.9 References and Further Reading
• 15 Fast Fourier Transforms (FFTs)
  • 15.1 Mathematical Definitions
  • 15.2 Overview of complex data FFTs
  • 15.3 Radix-2 FFT routines for complex data
  • 15.4 Mixed-radix FFT routines for complex data
  • 15.5 Overview of real data FFTs
  • 15.6 Radix-2 FFT routines for real data
  • 15.7 Mixed-radix FFT routines for real data
  • 15.8 References and Further Reading
• 16 Numerical Integration
  • 16.1 Introduction
    • 16.1.1 Integrands without weight functions
    • 16.1.2 Integrands with weight functions
    • 16.1.3 Integrands with singular weight functions
  • 16.2 QNG non-adaptive Gauss-Kronrod integration
  • 16.3 QAG adaptive integration
  • 16.4 QAGS adaptive integration with singularities
  • 16.5 QAGP adaptive integration with known singular points
  • 16.6 QAGI adaptive integration on infinite intervals
  • 16.7 QAWC adaptive integration for Cauchy principal values
  • 16.8 QAWS adaptive integration for singular functions
  • 16.9 QAWO adaptive integration for oscillatory functions
  • 16.10 QAWF adaptive integration for Fourier integrals
  • 16.11 Error codes
  • 16.12 Examples
  • 16.13 References and Further Reading
• 17 Random Number Generation
  • 17.1 General comments on random numbers
  • 17.2 The Random Number Generator Interface
  • 17.3 Random number generator initialization
  • 17.4 Sampling from a random number generator
  • 17.5 Auxiliary random number generator functions
  • 17.6 Random number environment variables
  • 17.7 Copying random number generator state
  • 17.8 Reading and writing random number generator state
  • 17.9 Random number generator algorithms
  • 17.10 Unix random number generators
  • 17.11 Other random number generators
  • 17.12 Performance
  • 17.13 Examples
  • 17.14 References and Further Reading
  • 17.15 Acknowledgements
• 18 Quasi-Random Sequences
  • 18.1 Quasi-random number generator initialization
  • 18.2 Sampling from a quasi-random number generator
  • 18.3 Auxiliary quasi-random number generator functions
  • 18.4 Saving and resorting quasi-random number generator state
  • 18.5 Quasi-random number generator algorithms
  • 18.6 Examples
  • 18.7 References
• 19 Random Number Distributions
  • 19.1 Introduction
  • 19.2 The Gaussian Distribution
  • 19.3 The Gaussian Tail Distribution
  • 19.4 The Bivariate Gaussian Distribution
  • 19.5 The Exponential Distribution
  • 19.6 The Laplace Distribution
  • 19.7 The Exponential Power Distribution
  • 19.8 The Cauchy Distribution
  • 19.9 The Rayleigh Distribution
  • 19.10 The Rayleigh Tail Distribution
  • 19.11 The Landau Distribution
  • 19.12 The Levy alpha-Stable Distributions
  • 19.13 The Levy skew alpha-Stable Distribution
  • 19.14 The Gamma Distribution
  • 19.15 The Flat (Uniform) Distribution
  • 19.16 The Lognormal Distribution
  • 19.17 The Chi-squared Distribution
  • 19.18 The F-distribution
  • 19.19 The t-distribution
  • 19.20 The Beta Distribution
  • 19.21 The Logistic Distribution
  • 19.22 The Pareto Distribution
  • 19.23 Spherical Vector Distributions
  • 19.24 The Weibull Distribution
  • 19.25 The Type-1 Gumbel Distribution
  • 19.26 The Type-2 Gumbel Distribution
  • 19.27 The Dirichlet Distribution
  • 19.28 General Discrete Distributions
  • 19.29 The Poisson Distribution
  • 19.30 The Bernoulli Distribution
  • 19.31 The Binomial Distribution
  • 19.32 The Multinomial Distribution
  • 19.33 The Negative Binomial Distribution
  • 19.34 The Pascal Distribution
  • 19.35 The Geometric Distribution
  • 19.36 The Hypergeometric Distribution
  • 19.37 The Logarithmic Distribution
  • 19.38 Shuffling and Sampling
  • 19.39 Examples
  • 19.40 References and Further Reading
• 20 Statistics
  • 20.1 Mean, Standard Deviation and Variance
  • 20.2 Absolute deviation
  • 20.3 Higher moments (skewness and kurtosis)
  • 20.4 Autocorrelation
  • 20.5 Covariance
  • 20.6 Correlation
  • 20.7 Weighted Samples
  • 20.8 Maximum and Minimum values
  • 20.9 Median and Percentiles
  • 20.10 Examples
  • 20.11 References and Further Reading
• 21 Histograms
  • 21.1 The histogram struct
  • 21.2 Histogram allocation
  • 21.3 Copying Histograms
  • 21.4 Updating and accessing histogram elements
  • 21.5 Searching histogram ranges
  • 21.6 Histogram Statistics
  • 21.7 Histogram Operations
  • 21.8 Reading and writing histograms
  • 21.9 Resampling from histograms
  • 21.10 The histogram probability distribution struct
  • 21.11 Example programs for histograms
  • 21.12 Two dimensional histograms
  • 21.13 The 2D histogram struct
  • 21.14 2D Histogram allocation
  • 21.15 Copying 2D Histograms
  • 21.16 Updating and accessing 2D histogram elements
  • 21.17 Searching 2D histogram ranges
  • 21.18 2D Histogram Statistics
  • 21.19 2D Histogram Operations
  • 21.20 Reading and writing 2D histograms
  • 21.21 Resampling from 2D histograms
  • 21.22 Example programs for 2D histograms
• 22 N-tuples
  • 22.1 The ntuple struct
  • 22.2 Creating ntuples
  • 22.3 Opening an existing ntuple file
  • 22.4 Writing ntuples
  • 22.5 Reading ntuples
  • 22.6 Closing an ntuple file
  • 22.7 Histogramming ntuple values
  • 22.8 Examples
  • 22.9 References and Further Reading
• 23 Monte Carlo Integration
  • 23.1 Interface
  • 23.2 PLAIN Monte Carlo
  • 23.3 MISER
  • 23.4 VEGAS
  • 23.5 Examples
  • 23.6 References and Further Reading
• 24 Simulated Annealing
  • 24.1 Simulated Annealing algorithm
  • 24.2 Simulated Annealing functions
  • 24.3 Examples
    • 24.3.1 Trivial example
    • 24.3.2 Traveling Salesman Problem
  • 24.4 References and Further Reading
• 25 Ordinary Differential Equations
  • 25.1 Defining the ODE System
  • 25.2 Stepping Functions
  • 25.3 Adaptive Step-size Control
  • 25.4 Evolution
  • 25.5 Examples
  • 25.6 References and Further Reading
• 26 Interpolation
  • 26.1 Introduction
  • 26.2 Interpolation Functions
  • 26.3 Interpolation Types
  • 26.4 Index Look-up and Acceleration
  • 26.5 Evaluation of Interpolating Functions
  • 26.6 Higher-level Interface
  • 26.7 Examples
  • 26.8 References and Further Reading
• 27 Numerical Differentiation
  • 27.1 Functions
  • 27.2 Examples
  • 27.3 References and Further Reading
• 28 Chebyshev Approximations
  • 28.1 Definitions
  • 28.2 Creation and Calculation of Chebyshev Series
  • 28.3 Auxiliary Functions
  • 28.4 Chebyshev Series Evaluation
  • 28.5 Derivatives and Integrals
  • 28.6 Examples
  • 28.7 References and Further Reading
• 29 Series Acceleration
  • 29.1 Acceleration functions
  • 29.2 Acceleration functions without error estimation
  • 29.3 Examples
  • 29.4 References and Further Reading
• 30 Wavelet Transforms
  • 30.1 Definitions
  • 30.2 Initialization
  • 30.3 Transform Functions
    • 30.3.1 Wavelet transforms in one dimension
    • 30.3.2 Wavelet transforms in two dimension
  • 30.4 Examples
  • 30.5 References and Further Reading
• 31 Discrete Hankel Transforms
  • 31.1 Definitions
  • 31.2 Functions
  • 31.3 References and Further Reading
• 32 One dimensional Root-Finding
  • 32.1 Overview
  • 32.2 Caveats
  • 32.3 Initializing the Solver
  • 32.4 Providing the function to solve
  • 32.5 Search Bounds and Guesses
  • 32.6 Iteration
  • 32.7 Search Stopping Parameters
  • 32.8 Root Bracketing Algorithms
  • 32.9 Root Finding Algorithms using Derivatives
  • 32.10 Examples
  • 32.11 References and Further Reading
• 33 One dimensional Minimization
  • 33.1 Overview
  • 33.2 Caveats
  • 33.3 Initializing the Minimizer
  • 33.4 Providing the function to minimize
  • 33.5 Iteration
  • 33.6 Stopping Parameters
  • 33.7 Minimization Algorithms
  • 33.8 Examples
  • 33.9 References and Further Reading
• 34 Multidimensional Root-Finding
  • 34.1 Overview
  • 34.2 Initializing the Solver
  • 34.3 Providing the function to solve
  • 34.4 Iteration
  • 34.5 Search Stopping Parameters
  • 34.6 Algorithms using Derivatives
  • 34.7 Algorithms without Derivatives
  • 34.8 Examples
  • 34.9 References and Further Reading
• 35 Multidimensional Minimization
  • 35.1 Overview
  • 35.2 Caveats
  • 35.3 Initializing the Multidimensional Minimizer
  • 35.4 Providing a function to minimize
  • 35.5 Iteration
  • 35.6 Stopping Criteria
  • 35.7 Algorithms with Derivatives
  • 35.8 Algorithms without Derivatives
  • 35.9 Examples
  • 35.10 References and Further Reading
• 36 Least-Squares Fitting
  • 36.1 Overview
  • 36.2 Linear regression
  • 36.3 Linear fitting without a constant term
  • 36.4 Multi-parameter fitting
  • 36.5 Examples
  • 36.6 References and Further Reading
• 37 Nonlinear Least-Squares Fitting
  • 37.1 Overview
  • 37.2 Initializing the Solver
  • 37.3 Providing the Function to be Minimized
  • 37.4 Iteration
  • 37.5 Search Stopping Parameters
  • 37.6 Minimization Algorithms using Derivatives
  • 37.7 Minimization Algorithms without Derivatives
  • 37.8 Computing the covariance matrix of best fit parameters
  • 37.9 Examples
  • 37.10 References and Further Reading
• 38 Basis Splines
  • 38.1 Overview
  • 38.2 Initializing the B-splines solver
  • 38.3 Constructing the knots vector
  • 38.4 Evaluation of B-splines
  • 38.5 Evaluation of B-spline derivatives
  • 38.6 Examples
  • 38.7 References and Further Reading
• 39 Physical Constants
  • 39.1 Fundamental Constants
  • 39.2 Astronomy and Astrophysics
  • 39.3 Atomic and Nuclear Physics
  • 39.4 Measurement of Time
  • 39.5 Imperial Units
  • 39.6 Speed and Nautical Units
  • 39.7 Printers Units
  • 39.8 Volume, Area and Length
  • 39.9 Mass and Weight
  • 39.10 Thermal Energy and Power
  • 39.11 Pressure
  • 39.12 Viscosity
  • 39.13 Light and Illumination
  • 39.14 Radioactivity
  • 39.15 Force and Energy
  • 39.16 Prefixes
  • 39.17 Examples
  • 39.18 References and Further Reading
• 40 IEEE floating-point arithmetic
  • 40.1 Representation of floating point numbers
  • 40.2 Setting up your IEEE environment
  • 40.3 References and Further Reading
• A Debugging Numerical Programs
  • 40.4 Using gdb
  • 40.5 Examining floating point registers
  • 40.6 Handling floating point exceptions
  • 40.7 GCC warning options for numerical programs
  • 40.8 References and Further Reading
• B Contributors to GSL
• C Autoconf Macros
• D GSL CBLAS Library
  • 40.9 Level 1
  • 40.10 Level 2
  • 40.11 Level 3
  • 40.12 Examples
• E GPG verification
• Free Software Needs Free Documentation
• Other books from the publisher
• GNU General Public License
• GNU Free Documentation License
• History
• Function Index
• Type and Variable Index
• Concept Index