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GNU Scientific Library
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GSL 2.6, Aug 16, 2019
The GSL Team
Copyright © 1996-2019 The GSL Team
Introduction Using the Library Error Handling Mathematical Functions Complex Numbers Polynomials Special Functions Vectors and Matrices Permutations Combinations Multisets Sorting BLAS Support Linear Algebra Eigensystems Fast Fourier Transforms (FFTs) Numerical Integration Random Number Generation Quasi-Random Sequences Random Number Distributions Statistics Running Statistics Moving Window Statistics Digital Filtering Histograms N-tuples Monte Carlo Integration Simulated Annealing Ordinary Differential Equations Interpolation Numerical Differentiation Chebyshev Approximations Series Acceleration Wavelet Transforms Discrete Hankel Transforms One Dimensional Root-Finding One Dimensional Minimization Multidimensional Root-Finding Multidimensional Minimization Linear Least-Squares Fitting Nonlinear Least-Squares Fitting Basis Splines Sparse Matrices Sparse BLAS Support Sparse Linear Algebra Physical Constants IEEE floating-point arithmetic Debugging Numerical Programs Contributors to GSL Autoconf Macros GSL CBLAS Library GNU General Public License GNU Free Documentation License Index— The Detailed Node Listing — Introduction
Routines available in GSL GSL is Free Software Obtaining GSL No Warranty Reporting Bugs Further Information Conventions used in this manualUsing 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 Deprecated Functions Code ReuseCompiling and Linking
Error Handling
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 NumbersComplex Numbers
Representation of 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 ReadingPolynomials
Polynomial Evaluation Divided Difference Representation of Polynomials Quadratic Equations Cubic Equations General Polynomial Equations Examples References and Further ReadingSpecial Functions
Usage The gsl_sf_result struct Modes Airy Functions and Derivatives Bessel Functions Clausen Functions Coulomb Functions Coupling Coefficients Dawson Function Debye Functions Dilogarithm Elementary Operations Elliptic Integrals Elliptic Functions (Jacobi) Error Functions Exponential Functions Exponential Integrals Fermi-Dirac Function Gamma and Beta Functions Gegenbauer Functions Hermite Polynomials and Functions Hypergeometric Functions Laguerre Functions Lambert W Functions Legendre Functions and Spherical Harmonics Logarithm and Related Functions Mathieu Functions Power Function Psi (Digamma) Function Synchrotron Functions Transport Functions Trigonometric Functions Zeta Functions Examples References and Further ReadingAiry Functions and Derivatives
Airy Functions Derivatives of Airy Functions Zeros of Airy Functions Zeros of Derivatives of Airy FunctionsBessel 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 FunctionsCoulomb Functions
Normalized Hydrogenic Bound States Coulomb Wave Functions Coulomb Wave Function Normalization ConstantCoupling Coefficients
Dilogarithm
Elliptic Integrals
Definition of Legendre Forms Definition of Carlson Forms Legendre Form of Complete Elliptic Integrals Legendre Form of Incomplete Elliptic Integrals Carlson FormsError Functions
Exponential Functions
Exponential Integrals
Exponential Integral Ei(x) Hyperbolic Integrals Ei_3(x) Trigonometric Integrals Arctangent IntegralFermi-Dirac Function
Gamma and Beta Functions
Gamma Functions Factorials Pochhammer Symbol Incomplete Gamma Functions Beta Functions Incomplete Beta FunctionHermite Polynomials and Functions
Hermite Polynomials Derivatives of Hermite Polynomials Hermite Functions Derivatives of Hermite Functions Zeros of Hermite Polynomials and Hermite FunctionsLegendre Functions and Spherical Harmonics
Legendre Polynomials Associated Legendre Polynomials and Spherical Harmonics Conical Functions Radial Functions for Hyperbolic SpaceMathieu Functions
Mathieu Function Workspace Mathieu Function Characteristic Values Angular Mathieu Functions Radial Mathieu FunctionsPsi (Digamma) Function
Trigonometric Functions
Circular Trigonometric Functions Trigonometric Functions for Complex Arguments Hyperbolic Trigonometric Functions Conversion Functions Restriction Functions Trigonometric Functions With Error EstimatesZeta Functions
Vectors and Matrices
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 vectorsMatrices
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 ReadingPermutations
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 ReadingCombinations
The Combination struct Combination allocation Accessing combination elements Combination properties Combination functions Reading and writing combinations Examples References and Further ReadingMultisets
The Multiset struct Multiset allocation Accessing multiset elements Multiset properties Multiset functions Reading and writing multisets ExamplesSorting
Sorting objects Sorting vectors Selecting the k smallest or largest elements Computing the rank Examples References and Further ReadingBLAS Support
GSL BLAS Interface Examples References and Further ReadingGSL BLAS Interface
Linear Algebra
LU Decomposition QR Decomposition QR Decomposition with Column Pivoting LQ Decomposition Complete Orthogonal Decomposition Singular Value Decomposition Cholesky Decomposition Pivoted Cholesky Decomposition Modified Cholesky Decomposition LDLT Decomposition Tridiagonal Decomposition of Real Symmetric Matrices Tridiagonal Decomposition of Hermitian Matrices Hessenberg Decomposition of Real Matrices Hessenberg-Triangular Decomposition of Real Matrices Bidiagonalization Givens Rotations Householder Transformations Householder solver for linear systems Tridiagonal Systems Triangular Systems Banded Systems Balancing Examples References and Further ReadingQR Decomposition
Banded Systems
General Banded Format Symmetric Banded Format Banded Cholesky Decomposition Banded LDLT DecompositionEigensystems
Real Symmetric Matrices Complex Hermitian Matrices Real Nonsymmetric Matrices Real Generalized Symmetric-Definite Eigensystems Complex Generalized Hermitian-Definite Eigensystems Real Generalized Nonsymmetric Eigensystems Sorting Eigenvalues and Eigenvectors Examples References and Further ReadingFast 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 ReadingNumerical 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 CQUAD doubly-adaptive integration Romberg integration Gauss-Legendre integration Fixed point quadratures Error codes Examples References and Further ReadingIntroduction
Integrands without weight functions Integrands with weight functions Integrands with singular weight functionsExamples
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 AcknowledgementsQuasi-Random Sequences
Quasi-random number generator initialization Sampling from a quasi-random number generator Auxiliary quasi-random number generator functions Saving and restoring quasi-random number generator state Quasi-random number generator algorithms Examples ReferencesRandom Number Distributions
Introduction The Gaussian Distribution The Gaussian Tail Distribution The Bivariate Gaussian Distribution The Multivariate 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 The Wishart Distribution Shuffling and Sampling Examples References and Further ReadingStatistics
Mean, Standard Deviation and Variance Absolute deviation Higher moments (skewness and kurtosis) Autocorrelation Covariance Correlation Weighted Samples Maximum and Minimum values Median and Percentiles Order Statistics Robust Location Estimates Robust Scale Estimates Examples References and Further ReadingRobust Location Estimates
Robust Scale Estimates
Median Absolute Deviation (MAD) S_n Statistic Q_n StatisticRunning Statistics
Initializing the Accumulator Adding Data to the Accumulator Current Statistics Quantiles Examples References and Further ReadingMoving Window Statistics
Introduction Handling Endpoints Allocation for Moving Window Statistics Moving Mean Moving Variance and Standard Deviation Moving Minimum and Maximum Moving Sum Moving Median Robust Scale Estimation User-defined Moving Statistics Accumulators Examples References and Further ReadingRobust Scale Estimation
Examples
Example 1 Example 2; Robust Scale Example 3; User-defined Moving WindowDigital Filtering
Introduction Handling Endpoints Linear Digital Filters Nonlinear Digital Filters Examples References and Further ReadingLinear Digital Filters
Nonlinear Digital Filters
Examples
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 histogramsN-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 ReadingMonte Carlo Integration
Interface PLAIN Monte Carlo MISER VEGAS Examples References and Further ReadingSimulated Annealing
Simulated Annealing algorithm Simulated Annealing functions Examples References and Further ReadingExamples
Ordinary Differential Equations
Defining the ODE System Stepping Functions Adaptive Step-size Control Evolution Driver Examples References and Further ReadingInterpolation
Introduction to 1D Interpolation 1D Interpolation Functions 1D Interpolation Types 1D Index Look-up and Acceleration 1D Evaluation of Interpolating Functions 1D Higher-level Interface 1D Interpolation Example Programs Introduction to 2D Interpolation 2D Interpolation Functions 2D Interpolation Grids 2D Interpolation Types 2D Evaluation of Interpolating Functions 2D Higher-level Interface 2D Interpolation Example programs References and Further ReadingNumerical Differentiation
Functions Examples References and Further ReadingChebyshev Approximations
Definitions Creation and Calculation of Chebyshev Series Auxiliary Functions Chebyshev Series Evaluation Derivatives and Integrals Examples References and Further ReadingSeries Acceleration
Acceleration functions Acceleration functions without error estimation Examples References and Further ReadingWavelet Transforms
Definitions Initialization Transform Functions Examples References and Further ReadingTransform Functions
Discrete Hankel Transforms
Definitions Functions References and Further ReadingOne 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 ReadingOne Dimensional Minimization
Overview Caveats Initializing the Minimizer Providing the function to minimize Iteration Stopping Parameters Minimization Algorithms Examples References and Further ReadingMultidimensional 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 ReadingMultidimensional Minimization
Overview Caveats Initializing the Multidimensional Minimizer Providing a function to minimize Iteration Stopping Criteria Algorithms with Derivatives Algorithms without Derivatives Examples References and Further ReadingLinear Least-Squares Fitting
Overview Linear regression Multi-parameter regression Regularized regression Robust linear regression Large dense linear systems Troubleshooting Examples References and Further ReadingLinear regression
Large dense linear systems
Normal Equations Approach Tall Skinny QR (TSQR) Approach Large Dense Linear Systems Solution Steps Large Dense Linear Least Squares RoutinesExamples
Simple Linear Regression Example Multi-parameter Linear Regression Example Regularized Linear Regression Example 1 Regularized Linear Regression Example 2 Robust Linear Regression Example Large Dense Linear Regression ExampleNonlinear Least-Squares Fitting
Overview Solving the Trust Region Subproblem (TRS) Weighted Nonlinear Least-Squares Tunable Parameters Initializing the Solver Providing the Function to be Minimized Iteration Testing for Convergence High Level Driver Covariance matrix of best fit parameters Troubleshooting Examples References and Further ReadingSolving the Trust Region Subproblem (TRS)
Levenberg-Marquardt Levenberg-Marquardt with Geodesic Acceleration Dogleg Double Dogleg Two Dimensional Subspace Steihaug-Toint Conjugate GradientExamples
Exponential Fitting Example Geodesic Acceleration Example 1 Geodesic Acceleration Example 2 Comparing TRS Methods Example Large Nonlinear Least Squares ExampleBasis Splines
Overview Initializing the B-splines solver Constructing the knots vector Evaluation of B-splines Evaluation of B-spline derivatives Working with the Greville abscissae Examples References and Further ReadingSparse Matrices
Data types Sparse Matrix Storage Formats Overview Allocation Accessing Matrix Elements Initializing Matrix Elements Reading and Writing Matrices Copying Matrices Exchanging Rows and Columns Matrix Operations Matrix Properties Finding Maximum and Minimum Elements Compressed Format Conversion Between Sparse and Dense Matrices Examples References and Further ReadingSparse Matrix Storage Formats
Coordinate Storage (COO) Compressed Sparse Column (CSC) Compressed Sparse Row (CSR)Sparse BLAS Support
Sparse BLAS operations References and Further ReadingSparse Linear Algebra
Overview Sparse Iterative Solvers Examples References and Further ReadingSparse Iterative Solvers
Physical Constants
Fundamental Constants Astronomy and Astrophysics Atomic and Nuclear Physics Measurement of Time Imperial Units Speed and Nautical Units Printers Units Volume, Area and Length Mass and Weight Thermal Energy and Power Pressure Viscosity Light and Illumination Radioactivity Force and Energy Prefixes Examples References and Further ReadingIEEE floating-point arithmetic
Representation of floating point numbers Setting up your IEEE environment References and Further ReadingDebugging Numerical Programs
Using gdb Examining floating point registers Handling floating point exceptions GCC warning options for numerical programs References and Further ReadingGSL CBLAS Library