GNU Scientific Library

Reference Manual

Edition 1.6, for GSL Version 1.6

27 December 2004

This file documents the GNU Scientific Library (GSL), a collection of numerical routines for scientific computing. It corresponds to release 1.6 of the library.

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

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License.

Printed copies of this manual can be purchased from Network Theory Ltd at http://www.network-theory.co.uk/gsl/manual/.

• Introduction
  • Routines available in GSL
  • GSL is Free Software
  • Obtaining GSL
  • No Warranty
  • Reporting Bugs
  • Further Information
  • Conventions used in this manual
• Using the library
  • An Example Program
  • Compiling and Linking
  • Shared Libraries
  • ANSI C Compliance
  • Inline functions
  • Long double
  • Portability functions
  • Alternative optimized functions
  • Support for different numeric types
  • Compatibility with C++
  • Aliasing of arrays
  • Thread-safety
  • Code Reuse
• Error Handling
  • Error Reporting
  • Error Codes
  • Error Handlers
  • Using GSL error reporting in your own functions
  • Examples
• Mathematical Functions
  • Mathematical Constants
  • Infinities and Not-a-number
  • Elementary Functions
  • Small integer powers
  • Testing the Sign of Numbers
  • Testing for Odd and Even Numbers
  • Maximum and Minimum functions
  • Approximate Comparison of Floating Point Numbers
• Complex Numbers
  • Complex numbers
  • Properties of complex numbers
  • Complex arithmetic operators
  • Elementary Complex Functions
  • Complex Trigonometric Functions
  • Inverse Complex Trigonometric Functions
  • Complex Hyperbolic Functions
  • Inverse Complex Hyperbolic Functions
  • References and Further Reading
• Polynomials
  • Polynomial Evaluation
  • Divided Difference Representation of Polynomials
  • Quadratic Equations
  • Cubic Equations
  • General Polynomial Equations
  • Examples
  • References and Further Reading
• Special Functions
  • Usage
  • The gsl_sf_result struct
  • Modes
  • Airy Functions and Derivatives
    • Airy Functions
    • Derivatives of Airy Functions
    • Zeros of Airy Functions
    • Zeros of Derivatives of Airy Functions
  • Bessel Functions
    • Regular Cylindrical Bessel Functions
    • Irregular Cylindrical Bessel Functions
    • Regular Modified Cylindrical Bessel Functions
    • Irregular Modified Cylindrical Bessel Functions
    • Regular Spherical Bessel Functions
    • Irregular Spherical Bessel Functions
    • Regular Modified Spherical Bessel Functions
    • Irregular Modified Spherical Bessel Functions
    • Regular Bessel Function - Fractional Order
    • Irregular Bessel Functions - Fractional Order
    • Regular Modified Bessel Functions - Fractional Order
    • Irregular Modified Bessel Functions - Fractional Order
    • Zeros of Regular Bessel Functions
  • Clausen Functions
  • Coulomb Functions
    • Normalized Hydrogenic Bound States
    • Coulomb Wave Functions
    • Coulomb Wave Function Normalization Constant
  • Coupling Coefficients
    • 3-j Symbols
    • 6-j Symbols
    • 9-j Symbols
  • Dawson Function
  • Debye Functions
  • Dilogarithm
    • Real Argument
    • Complex Argument
  • Elementary Operations
  • Elliptic Integrals
    • Definition of Legendre Forms
    • Definition of Carlson Forms
    • Legendre Form of Complete Elliptic Integrals
    • Legendre Form of Incomplete Elliptic Integrals
    • Carlson Forms
  • Elliptic Functions (Jacobi)
  • Error Functions
    • Error Function
    • Complementary Error Function
    • Log Complementary Error Function
    • Probability functions
  • Exponential Functions
    • Exponential Function
    • Relative Exponential Functions
    • Exponentiation With Error Estimate
  • Exponential Integrals
    • Exponential Integral
    • Ei(x)
    • Hyperbolic Integrals
    • Ei_3(x)
    • Trigonometric Integrals
    • Arctangent Integral
  • Fermi-Dirac Function
    • Complete Fermi-Dirac Integrals
    • Incomplete Fermi-Dirac Integrals
  • Gamma Function
  • Gegenbauer Functions
  • Hypergeometric Functions
  • Laguerre Functions
  • Lambert W Functions
  • Legendre Functions and Spherical Harmonics
    • Legendre Polynomials
    • Associated Legendre Polynomials and Spherical Harmonics
    • Conical Functions
    • Radial Functions for Hyperbolic Space
  • Logarithm and Related Functions
  • Power Function
  • Psi (Digamma) Function
    • Digamma Function
    • Trigamma Function
    • Polygamma Function
  • Synchrotron Functions
  • Transport Functions
  • Trigonometric Functions
    • Circular Trigonometric Functions
    • Trigonometric Functions for Complex Arguments
    • Hyperbolic Trigonometric Functions
    • Conversion Functions
    • Restriction Functions
    • Trigonometric Functions With Error Estimates
  • Zeta Functions
    • Riemann Zeta Function
    • Riemann Zeta Function Minus One
    • Hurwitz Zeta Function
    • Eta Function
  • Examples
  • References and Further Reading
• Vectors and Matrices
  • Data types
  • Blocks
    • Block allocation
    • Reading and writing blocks
    • Example programs for blocks
  • Vectors
    • Vector allocation
    • Accessing vector elements
    • Initializing vector elements
    • Reading and writing vectors
    • Vector views
    • Copying vectors
    • Exchanging elements
    • Vector operations
    • Finding maximum and minimum elements of vectors
    • Vector properties
    • Example programs for vectors
  • Matrices
    • Matrix allocation
    • Accessing matrix elements
    • Initializing matrix elements
    • Reading and writing matrices
    • Matrix views
    • Creating row and column views
    • Copying matrices
    • Copying rows and columns
    • Exchanging rows and columns
    • Matrix operations
    • Finding maximum and minimum elements of matrices
    • Matrix properties
    • Example programs for matrices
  • References and Further Reading
• Permutations
  • The Permutation struct
  • Permutation allocation
  • Accessing permutation elements
  • Permutation properties
  • Permutation functions
  • Applying Permutations
  • Reading and writing permutations
  • Permutations in Cyclic Form
  • Examples
  • References and Further Reading
• Combinations
  • The Combination struct
  • Combination allocation
  • Accessing combination elements
  • Combination properties
  • Combination functions
  • Reading and writing combinations
  • Examples
  • References and Further Reading
• Sorting
  • Sorting objects
  • Sorting vectors
  • Selecting the k smallest or largest elements
  • Computing the rank
  • Examples
  • References and Further Reading
• BLAS Support
  • GSL BLAS Interface
    • Level 1
    • Level 2
    • Level 3
  • Examples
  • References and Further Reading
• Linear Algebra
  • LU Decomposition
  • QR Decomposition
  • QR Decomposition with Column Pivoting
  • Singular Value Decomposition
  • Cholesky Decomposition
  • Tridiagonal Decomposition of Real Symmetric Matrices
  • Tridiagonal Decomposition of Hermitian Matrices
  • Bidiagonalization
  • Householder Transformations
  • Householder solver for linear systems
  • Tridiagonal Systems
  • Examples
  • References and Further Reading
• Eigensystems
  • Real Symmetric Matrices
  • Complex Hermitian Matrices
  • Sorting Eigenvalues and Eigenvectors
  • Examples
  • References and Further Reading
• Fast Fourier Transforms (FFTs)
  • Mathematical Definitions
  • Overview of complex data FFTs
  • Radix-2 FFT routines for complex data
  • Mixed-radix FFT routines for complex data
  • Overview of real data FFTs
  • Radix-2 FFT routines for real data
  • Mixed-radix FFT routines for real data
  • References and Further Reading
• Numerical Integration
  • Introduction
  • QNG non-adaptive Gauss-Kronrod integration
  • QAG adaptive integration
  • QAGS adaptive integration with singularities
  • QAGP adaptive integration with known singular points
  • QAGI adaptive integration on infinite intervals
  • QAWC adaptive integration for Cauchy principal values
  • QAWS adaptive integration for singular functions
  • QAWO adaptive integration for oscillatory functions
  • QAWF adaptive integration for Fourier integrals
  • Error codes
  • Examples
  • References and Further Reading
• Random Number Generation
  • General comments on random numbers
  • The Random Number Generator Interface
  • Random number generator initialization
  • Sampling from a random number generator
  • Auxiliary random number generator functions
  • Random number environment variables
  • Copying random number generator state
  • Reading and writing random number generator state
  • Random number generator algorithms
  • Unix random number generators
  • Other random number generators
  • Performance
  • Examples
  • References and Further Reading
  • Acknowledgements
• Quasi-Random Sequences
  • Quasi-random number generator initialization
  • Sampling from a quasi-random number generator
  • Auxiliary quasi-random number generator functions
  • Saving and resorting quasi-random number generator state
  • Quasi-random number generator algorithms
  • Examples
  • References
• Random Number Distributions
  • Introduction
  • The Gaussian Distribution
  • The Gaussian Tail Distribution
  • The Bivariate Gaussian Distribution
  • The Exponential Distribution
  • The Laplace Distribution
  • The Exponential Power Distribution
  • The Cauchy Distribution
  • The Rayleigh Distribution
  • The Rayleigh Tail Distribution
  • The Landau Distribution
  • The Levy alpha-Stable Distributions
  • The Levy skew alpha-Stable Distribution
  • The Gamma Distribution
  • The Flat (Uniform) Distribution
  • The Lognormal Distribution
  • The Chi-squared Distribution
  • The F-distribution
  • The t-distribution
  • The Beta Distribution
  • The Logistic Distribution
  • The Pareto Distribution
  • Spherical Vector Distributions
  • The Weibull Distribution
  • The Type-1 Gumbel Distribution
  • The Type-2 Gumbel Distribution
  • The Dirichlet Distribution
  • General Discrete Distributions
  • The Poisson Distribution
  • The Bernoulli Distribution
  • The Binomial Distribution
  • The Multinomial Distribution
  • The Negative Binomial Distribution
  • The Pascal Distribution
  • The Geometric Distribution
  • The Hypergeometric Distribution
  • The Logarithmic Distribution
  • Shuffling and Sampling
  • Examples
  • References and Further Reading
• Statistics
  • Mean, Standard Deviation and Variance
  • Absolute deviation
  • Higher moments (skewness and kurtosis)
  • Autocorrelation
  • Covariance
  • Weighted Samples
  • Maximum and Minimum values
  • Median and Percentiles
  • Examples
  • References and Further Reading
• Histograms
  • The histogram struct
  • Histogram allocation
  • Copying Histograms
  • Updating and accessing histogram elements
  • Searching histogram ranges
  • Histogram Statistics
  • Histogram Operations
  • Reading and writing histograms
  • Resampling from histograms
  • The histogram probability distribution struct
  • Example programs for histograms
  • Two dimensional histograms
  • The 2D histogram struct
  • 2D Histogram allocation
  • Copying 2D Histograms
  • Updating and accessing 2D histogram elements
  • Searching 2D histogram ranges
  • 2D Histogram Statistics
  • 2D Histogram Operations
  • Reading and writing 2D histograms
  • Resampling from 2D histograms
  • Example programs for 2D histograms
• N-tuples
  • The ntuple struct
  • Creating ntuples
  • Opening an existing ntuple file
  • Writing ntuples
  • Reading ntuples
  • Closing an ntuple file
  • Histogramming ntuple values
  • Examples
  • References and Further Reading
• Monte Carlo Integration
  • Interface
  • PLAIN Monte Carlo
  • MISER
  • VEGAS
  • Examples
  • References and Further Reading
• Simulated Annealing
  • Simulated Annealing algorithm
  • Simulated Annealing functions
  • Examples
    • Trivial example
    • Traveling Salesman Problem
• Ordinary Differential Equations
  • Defining the ODE System
  • Stepping Functions
  • Adaptive Step-size Control
  • Evolution
  • Examples
  • References and Further Reading
• Interpolation
  • Introduction
  • Interpolation Functions
  • Interpolation Types
  • Index Look-up and Acceleration
  • Evaluation of Interpolating Functions
  • Higher-level Interface
  • Examples
  • References and Further Reading
• Numerical Differentiation
  • Functions
  • Examples
  • References and Further Reading
• Chebyshev Approximations
  • The gsl_cheb_series struct
  • Creation and Calculation of Chebyshev Series
  • Chebyshev Series Evaluation
  • Derivatives and Integrals
  • Examples
  • References and Further Reading
• Series Acceleration
  • Acceleration functions
  • Acceleration functions without error estimation
  • Examples
  • References and Further Reading
• Wavelet Transforms
  • Definitions
  • Initialization
  • Transform Functions
    • Wavelet transforms in one dimension
    • Wavelet transforms in two dimension
  • Example
  • References and Further Reading
• Discrete Hankel Transforms
  • Definitions
  • Functions
  • References and Further Reading
• One dimensional Root-Finding
  • Overview
  • Caveats
  • Initializing the Solver
  • Providing the function to solve
  • Search Bounds and Guesses
  • Iteration
  • Search Stopping Parameters
  • Root Bracketing Algorithms
  • Root Finding Algorithms using Derivatives
  • Examples
  • References and Further Reading
• One dimensional Minimization
  • Overview
  • Caveats
  • Initializing the Minimizer
  • Providing the function to minimize
  • Iteration
  • Stopping Parameters
  • Minimization Algorithms
  • Examples
  • References and Further Reading
• Multidimensional Root-Finding
  • Overview
  • Initializing the Solver
  • Providing the function to solve
  • Iteration
  • Search Stopping Parameters
  • Algorithms using Derivatives
  • Algorithms without Derivatives
  • Examples
  • References and Further Reading
• Multidimensional Minimization
  • Overview
  • Caveats
  • Initializing the Multidimensional Minimizer
  • Providing a function to minimize
  • Iteration
  • Stopping Criteria
  • Algorithms
  • Examples
  • References and Further Reading
• Least-Squares Fitting
  • Linear regression
  • Linear fitting without a constant term
  • Multi-parameter fitting
  • Examples
  • References and Further Reading
• Nonlinear Least-Squares Fitting
  • Overview
  • Initializing the Solver
  • Providing the Function to be Minimized
  • Iteration
  • Search Stopping Parameters
  • Minimization Algorithms using Derivatives
  • Minimization Algorithms without Derivatives
  • Computing the covariance matrix of best fit parameters
  • Examples
  • References and Further Reading
• Physical Constants
  • Fundamental Constants
  • Astronomy and Astrophysics
  • Atomic and Nuclear Physics
  • Measurement of Time
  • Imperial Units
  • Nautical Units
  • Printers Units
  • Volume
  • Mass and Weight
  • Thermal Energy and Power
  • Pressure
  • Viscosity
  • Light and Illumination
  • Radioactivity
  • Force and Energy
  • Prefixes
  • Examples
  • References and Further Reading
• IEEE floating-point arithmetic
  • Representation of floating point numbers
  • Setting up your IEEE environment
  • References and Further Reading
• Debugging Numerical Programs
  • Using gdb
  • Examining floating point registers
  • Handling floating point exceptions
  • GCC warning options for numerical programs
  • References and Further Reading
• Contributors to GSL
• Autoconf Macros
• GSL CBLAS Library
  • Level 1
  • Level 2
  • Level 3
  • Examples
• Free Software Needs Free Documentation
• GNU General Public License
  • Preamble
  • Appendix: How to Apply These Terms to Your New Programs
• GNU Free Documentation License
  • ADDENDUM: How to use this License for your documents
• Function Index
• Variable Index
• Type Index
• Concept Index

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