GNU Scientific Library

Reference Manual

Edition 1.0, for GSL Version 1.0

1 November 2001

Mark Galassi

Jim Davies

James Theiler

Brian Gough

Gerard Jungman

Michael Booth

Fabrice Rossi

Introduction
- Routines available in GSL
- GSL is Free Software
- Obtaining GSL
- An Example Program
- No Warranty
- Further Information
Using the library
- ANSI C Compliance
- Compiling and Linking
- Shared Libraries
- Autoconf macros
- Inline functions
- Long double
- Portability functions
- Alternative optimized functions
- Support for different numeric types
- Compatibility with C++
- Aliasing of arrays
- Code Reuse
Error Handling
- Error Reporting
- Error Handlers
- Using GSL error reporting in your own functions
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
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
Roots of Polynomials
- Polynomial evaluation
- 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
  - Hurwitz Zeta Function
  - Eta Function
- Examples
- References and Further Reading
Vectors and Matrices
- Data types
- Blocks
- Vectors
- Matrices
- References and Further Reading
Permutations
- The Permutation struct
- Permutation allocation
- Accessing permutation elements
- Permutation properties
- Permutation functions
- Applying Permutations
- Reading and writing permutations
- Examples
- References and Further Reading
Sorting
- Sorting objects
- Sorting vectors
- Selecting the k-th smallest or largest elements
- Computing the rank
- Examples
- References and Further Reading
BLAS Support
- GSL BLAS Interface
- 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 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
- Saving and restoring random number generator state
- Random number generator algorithms
- Unix random number generators
- Numerical Recipes generators
- Other random number generators
- Random Number Generator Performance
- 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
- 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
- The Spherical Distribution (2D & 3D)
- The Weibull Distribution
- The Type-1 Gumbel Distribution
- The Type-2 Gumbel Distribution
- General Discrete Distributions
- The Poisson Distribution
- The Bernoulli Distribution
- The Binomial 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
- Example statistical programs
- 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
- Example programs
- 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 with Simulated Annealing
  - 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
- Example
- 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
- Example of accelerating a series
- 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
- 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
GNU General Public License
- Preamble
- TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
- 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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