This is gsl-ref.info, produced by makeinfo version 4.0 from gsl-ref.texi. INFO-DIR-SECTION Scientific software START-INFO-DIR-ENTRY * gsl-ref: (gsl-ref). GNU Scientific Library - Reference END-INFO-DIR-ENTRY This file documents the GNU Scientific Library. Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001 The GSL Team. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 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".  Indirect: gsl-ref.info-1: 720 gsl-ref.info-2: 50378 gsl-ref.info-3: 98880 gsl-ref.info-4: 147852 gsl-ref.info-5: 195208 gsl-ref.info-6: 244495 gsl-ref.info-7: 293518 gsl-ref.info-8: 343509 gsl-ref.info-9: 390132 gsl-ref.info-10: 432303 gsl-ref.info-11: 481661 gsl-ref.info-12: 530518 gsl-ref.info-13: 580187 gsl-ref.info-14: 629776 gsl-ref.info-15: 678117 gsl-ref.info-16: 728106 gsl-ref.info-17: 777046 gsl-ref.info-18: 826753 gsl-ref.info-19: 869693 gsl-ref.info-20: 916449 gsl-ref.info-21: 936366 gsl-ref.info-22: 1046435  Tag Table: (Indirect) Node: Top720 Node: Introduction2249 Node: Routines available in GSL2868 Node: GSL is Free Software4509 Node: Obtaining GSL6633 Node: An Example Program7811 Node: No Warranty8551 Node: Further Information8992 Node: Using the library9521 Node: ANSI C Compliance10060 Node: Compiling and Linking11122 Node: Shared Libraries13325 Node: Autoconf macros13892 Node: Inline functions16183 Node: Long double17507 Node: Portability functions18900 Node: Alternative optimized functions20357 Node: Support for different numeric types21808 Node: Compatibility with C++24870 Node: Aliasing of arrays25185 Node: Code Reuse25926 Node: Error Handling26561 Node: Error Reporting27265 Node: Error Handlers31052 Node: Using GSL error reporting in your own functions35230 Node: Mathematical Functions37141 Node: Mathematical Constants37848 Node: Infinities and Not-a-number38971 Node: Elementary Functions40025 Node: Small integer powers41715 Node: Testing the Sign of Numbers43291 Node: Testing for Odd and Even Numbers43713 Node: Maximum and Minimum functions44259 Node: Complex Numbers46268 Node: Complex numbers47612 Node: Properties of complex numbers49403 Node: Complex arithmetic operators50378 Node: Elementary Complex Functions53101 Node: Complex Trigonometric Functions54905 Node: Inverse Complex Trigonometric Functions56112 Node: Complex Hyperbolic Functions58698 Node: Inverse Complex Hyperbolic Functions59980 Node: Complex Number References and Further Reading61804 Node: Polynomials63193 Node: Polynomial Evaluation63960 Node: Divided Difference Representation of Polynomials64411 Node: Quadratic Equations66069 Node: Cubic Equations67836 Node: General Polynomial Equations69181 Node: Roots of Polynomials Examples71052 Node: Roots of Polynomials References and Further Reading72467 Node: Special Functions73227 Node: Special Function Usage75172 Node: The gsl_sf_result struct76351 Node: Special Function Modes77610 Node: Airy Functions and Derivatives78542 Node: Airy Functions79233 Node: Derivatives of Airy Functions80573 Node: Zeros of Airy Functions81912 Node: Zeros of Derivatives of Airy Functions82615 Node: Bessel Functions83358 Node: Regular Cylindrical Bessel Functions84545 Node: Irregular Cylindrical Bessel Functions85851 Node: Regular Modified Cylindrical Bessel Functions87289 Node: Irregular Modified Cylindrical Bessel Functions90092 Node: Regular Spherical Bessel Functions93036 Node: Irregular Spherical Bessel Functions95227 Node: Regular Modified Spherical Bessel Functions96894 Node: Irregular Modified Spherical Bessel Functions98880 Node: Regular Bessel Function - Fractional Order100926 Node: Irregular Bessel Functions - Fractional Order101910 Node: Regular Modified Bessel Functions - Fractional Order102477 Node: Irregular Modified Bessel Functions - Fractional Order103384 Node: Zeros of Regular Bessel Functions104599 Node: Clausen Functions105609 Node: Coulomb Functions106206 Node: Normalized Hydrogenic Bound States106632 Node: Coulomb Wave Functions107638 Node: Coulomb Wave Function Normalization Constant111023 Node: Coupling Coefficients111765 Node: 3-j Symbols112483 Node: 6-j Symbols113073 Node: 9-j Symbols113683 Node: Dawson Function114375 Node: Debye Functions114934 Node: Dilogarithm116214 Node: Real Argument116502 Node: Complex Argument117055 Node: Elementary Operations117509 Node: Elliptic Integrals118324 Node: Definition of Legendre Forms118790 Node: Definition of Carlson Forms119628 Node: Legendre Form of Complete Elliptic Integrals120329 Node: Legendre Form of Incomplete Elliptic Integrals121195 Node: Carlson Forms122912 Node: Elliptic Functions (Jacobi)124437 Node: Error Functions125010 Node: Error Function125453 Node: Complementary Error Function125797 Node: Log Complementary Error Function126242 Node: Probability functions126678 Node: Exponential Functions127417 Node: Exponential Function127793 Node: Relative Exponential Functions128986 Node: Exponentiation With Error Estimate130635 Node: Exponential Integrals131826 Node: Exponential Integral132301 Node: Ei(x)132928 Node: Hyperbolic Integrals133358 Node: Ei_3(x)133995 Node: Trigonometric Integrals134360 Node: Arctangent Integral134941 Node: Fermi-Dirac Function135304 Node: Complete Fermi-Dirac Integrals135657 Node: Incomplete Fermi-Dirac Integrals138088 Node: Gamma Function138715 Node: Gegenbauer Functions146228 Node: Hypergeometric Functions147852 Node: Laguerre Functions152056 Node: Lambert W Functions153310 Node: Legendre Functions and Spherical Harmonics154327 Node: Legendre Polynomials154909 Node: Associated Legendre Polynomials and Spherical Harmonics156726 Node: Conical Functions159134 Node: Radial Functions for Hyperbolic Space161231 Node: Logarithm and Related Functions163389 Node: Power Function164994 Node: Psi (Digamma) Function165926 Node: Digamma Function166430 Node: Trigamma Function167239 Node: Polygamma Function167607 Node: Synchrotron Functions167975 Node: Transport Functions168727 Node: Trigonometric Functions169819 Node: Circular Trigonometric Functions170434 Node: Trigonometric Functions for Complex Arguments171421 Node: Hyperbolic Trigonometric Functions172438 Node: Conversion Functions173023 Node: Restriction Functions173790 Node: Trigonometric Functions With Error Estimates174401 Node: Zeta Functions175145 Node: Riemann Zeta Function175560 Node: Hurwitz Zeta Function176214 Node: Eta Function176696 Node: Special Functions Examples177240 Node: Special Functions References and Further Reading178921 Node: Vectors and Matrices180017 Node: Data types180775 Node: Blocks181973 Node: Block allocation182884 Node: Reading and writing blocks184276 Node: Example programs for blocks186327 Node: Vectors186958 Node: Vector allocation188823 Node: Accessing vector elements190347 Node: Initializing vector elements192462 Node: Reading and writing vectors193141 Node: Vector views195208 Node: Copying vectors202389 Node: Exchanging elements203238 Node: Vector operations203785 Node: Finding maximum and minimum elements of vectors205284 Node: Vector properties206791 Node: Example programs for vectors207139 Node: Matrices208875 Node: Matrix allocation211728 Node: Accessing matrix elements213301 Node: Initializing matrix elements214983 Node: Reading and writing matrices215764 Node: Matrix views217844 Node: Creating row and column views224965 Node: Copying matrices228114 Node: Copying rows and columns228712 Node: Exchanging rows and columns230376 Node: Matrix operations231834 Node: Finding maximum and minimum elements of matrices233450 Node: Matrix properties235137 Node: Example programs for matrices235488 Node: Vector and Matrix References and Further Reading239401 Node: Permutations239879 Node: The Permutation struct241134 Node: Permutation allocation241630 Node: Accessing permutation elements243051 Node: Permutation properties243807 Node: Permutation functions244495 Node: Applying Permutations245740 Node: Reading and writing permutations247343 Node: Permutation Examples249508 Node: Permutation References and Further Reading251950 Node: Combinations252363 Node: The Combination struct253189 Node: Combination allocation253732 Node: Accessing combination elements255235 Node: Combination properties255798 Node: Combination functions256673 Node: Reading and writing combinations257602 Node: Combination Examples259781 Node: Sorting261058 Node: Sorting objects262004 Node: Sorting vectors264906 Node: Selecting the k-th smallest or largest elements267475 Node: Computing the rank270921 Node: Sorting Examples272129 Node: Sorting References and Further Reading273746 Node: BLAS Support274282 Node: GSL BLAS Interface277250 Node: Level 1 GSL BLAS Interface277730 Node: Level 2 GSL BLAS Interface284240 Node: Level 3 GSL BLAS Interface293518 Node: BLAS Examples304591 Node: BLAS References and Further Reading306042 Node: Linear Algebra307386 Node: LU Decomposition308593 Node: QR Decomposition313513 Node: QR Decomposition with Column Pivoting319155 Node: Singular Value Decomposition323527 Node: Cholesky Decomposition326824 Node: Tridiagonal Decomposition of Real Symmetric Matrices328725 Node: Tridiagonal Decomposition of Hermitian Matrices330563 Node: Bidiagonalization332481 Node: Householder solver for linear systems334793 Node: Tridiagonal Systems335674 Node: Linear Algebra Examples336769 Node: Linear Algebra References and Further Reading338797 Node: Eigensystems340254 Node: Real Symmetric Matrices341293 Node: Complex Hermitian Matrices343509 Node: Sorting Eigenvalues and Eigenvectors346011 Node: Eigenvalue and Eigenvector Examples347330 Node: Eigenvalue and Eigenvector References349680 Node: Fast Fourier Transforms350378 Node: Mathematical Definitions351584 Node: Overview of complex data FFTs353982 Node: Radix-2 FFT routines for complex data356601 Node: Mixed-radix FFT routines for complex data360666 Node: Overview of real data FFTs369911 Node: Radix-2 FFT routines for real data372341 Node: Mixed-radix FFT routines for real data375170 Node: FFT References and Further Reading385830 Node: Numerical Integration388590 Node: Numerical Integration Introduction390132 Node: QNG non-adaptive Gauss-Kronrod integration393595 Node: QAG adaptive integration394855 Node: QAGS adaptive integration with singularities397485 Node: QAGP adaptive integration with known singular points399275 Node: QAGI adaptive integration on infinite intervals400575 Node: QAWC adaptive integration for Cauchy principal values402797 Node: QAWS adaptive integration for singular functions403960 Node: QAWO adaptive integration for oscillatory functions407189 Node: QAWF adaptive integration for Fourier integrals410943 Node: Numerical integration error codes413521 Node: Numerical integration examples414280 Node: Numerical integration References and Further Reading416306 Node: Random Number Generation417022 Node: General comments on random numbers418510 Node: The Random Number Generator Interface420458 Node: Random number generator initialization421809 Node: Sampling from a random number generator423806 Node: Auxiliary random number generator functions426281 Node: Random number environment variables428601 Node: Saving and restoring random number generator state431134 Node: Random number generator algorithms432303 Node: Unix random number generators441165 Node: Numerical Recipes generators444880 Node: Other random number generators446263 Node: Random Number Generator Performance455552 Node: Random Number Generator Examples456831 Node: Random Number References and Further Reading458407 Node: Random Number Acknowledgements459601 Node: Quasi-Random Sequences460075 Node: Quasi-random number generator initialization461099 Node: Sampling from a quasi-random number generator462003 Node: Auxiliary quasi-random number generator functions462597 Node: Saving and resorting quasi-random number generator state463529 Node: Quasi-random number generator algorithms464315 Node: Quasi-random number generator examples465081 Node: Quasi-random number references466071 Node: Random Number Distributions466584 Node: The Gaussian Distribution468848 Node: The Gaussian Tail Distribution470431 Node: The Bivariate Gaussian Distribution472044 Node: The Exponential Distribution473323 Node: The Laplace Distribution474026 Node: The Exponential Power Distribution474715 Node: The Cauchy Distribution475668 Node: The Rayleigh Distribution476501 Node: The Rayleigh Tail Distribution477227 Node: The Landau Distribution478083 Node: The Levy alpha-Stable Distributions479021 Node: The Levy skew alpha-Stable Distribution480064 Node: The Gamma Distribution481661 Node: The Flat (Uniform) Distribution482371 Node: The Lognormal Distribution483075 Node: The Chi-squared Distribution483874 Node: The F-distribution484850 Node: The t-distribution485981 Node: The Beta Distribution486961 Node: The Logistic Distribution487661 Node: The Pareto Distribution488362 Node: The Spherical Distribution (2D & 3D)489065 Node: The Weibull Distribution491874 Node: The Type-1 Gumbel Distribution492605 Node: The Type-2 Gumbel Distribution493372 Node: General Discrete Distributions494137 Node: The Poisson Distribution497987 Node: The Bernoulli Distribution498702 Node: The Binomial Distribution499428 Node: The Negative Binomial Distribution500296 Node: The Pascal Distribution501297 Node: The Geometric Distribution502120 Node: The Hypergeometric Distribution502924 Node: The Logarithmic Distribution503889 Node: Shuffling and Sampling504665 Node: Random Number Distribution Examples507504 Node: Random Number Distribution References and Further Reading509903 Node: Statistics511106 Node: Mean and standard deviation and variance512383 Node: Absolute deviation515428 Node: Higher moments (skewness and kurtosis)516712 Node: Autocorrelation518839 Node: Covariance519642 Node: Weighted Samples520461 Node: Maximum and Minimum values525672 Node: Median and Percentiles528108 Node: Example statistical programs530518 Node: Statistics References and Further Reading533252 Node: Histograms534436 Node: The histogram struct536190 Node: Histogram allocation537990 Node: Copying Histograms540971 Node: Updating and accessing histogram elements541635 Node: Searching histogram ranges544884 Node: Histogram Statistics545875 Node: Histogram Operations547718 Node: Reading and writing histograms549761 Node: Resampling from histograms552782 Node: The histogram probability distribution struct553573 Node: Example programs for histograms556600 Node: Two dimensional histograms558721 Node: The 2D histogram struct559433 Node: 2D Histogram allocation561223 Node: Copying 2D Histograms563281 Node: Updating and accessing 2D histogram elements563972 Node: Searching 2D histogram ranges567598 Node: 2D Histogram Statistics568607 Node: 2D Histogram Operations571388 Node: Reading and writing 2D histograms573538 Node: Resampling from 2D histograms577182 Node: Example programs for 2D histograms580187 Node: N-tuples582245 Node: The ntuple struct583505 Node: Creating ntuples583976 Node: Opening an existing ntuple file584633 Node: Writing ntuples585251 Node: Reading ntuples585699 Node: Closing an ntuple file586018 Node: Histogramming ntuple values586347 Node: Example ntuple programs588343 Node: Ntuple References and Further Reading591895 Node: Monte Carlo Integration592206 Node: Monte Carlo Interface593450 Node: PLAIN Monte Carlo596084 Node: MISER598511 Node: VEGAS604334 Node: Monte Carlo Examples612631 Node: Monte Carlo Integration References and Further Reading618659 Node: Simulated Annealing619411 Node: Simulated Annealing algorithm620560 Node: Simulated Annealing functions622080 Node: Examples with Simulated Annealing626551 Node: Trivial example627089 Node: Traveling Salesman Problem629776 Node: Ordinary Differential Equations633194 Node: Defining the ODE System634012 Node: Stepping Functions635826 Node: Adaptive Step-size Control639381 Node: Evolution644071 Node: ODE Example programs646126 Node: ODE References and Further Reading650972 Node: Interpolation651605 Node: Introduction to Interpolation652615 Node: Interpolation Functions653045 Node: Interpolation Types654211 Node: Index Look-up and Acceleration656014 Node: Evaluation of interpolating functions657627 Node: Higher-level interface659979 Node: Interpolation Example programs661827 Node: Interpolation References and Further Reading663557 Node: Numerical Differentiation663987 Node: Numerical Differentiation functions664567 Node: Numerical Differentiation Example666174 Node: Numerical Differentiation References667586 Node: Chebyshev Approximations667984 Node: The gsl_cheb_series struct668940 Node: Creation and Calculation of Chebyshev Series669585 Node: Chebyshev Series Evaluation670549 Node: Derivatives and Integrals671909 Node: Chebyshev Approximation examples673142 Node: Chebyshev Approximation References and Further Reading674676 Node: Series Acceleration675115 Node: Acceleration functions675885 Node: Acceleration functions without error estimation678117 Node: Example of accelerating a series680723 Node: Series Acceleration References683164 Node: Discrete Hankel Transforms684043 Node: Discrete Hankel Transform Definition684509 Node: Discrete Hankel Transform Functions686347 Node: Discrete Hankel Transform References687851 Node: One dimensional Root-Finding688245 Node: Root Finding Overview689502 Node: Root Finding Caveats691353 Node: Initializing the Solver693111 Node: Providing the function to solve695728 Node: Search Bounds and Guesses699532 Node: Root Finding Iteration700386 Node: Search Stopping Parameters702222 Node: Root Bracketing Algorithms704730 Node: Root Finding Algorithms using Derivatives708019 Node: Root Finding Examples711490 Node: Root Finding References and Further Reading718838 Node: One dimensional Minimization719465 Node: Minimization Overview720764 Node: Minimization Caveats722438 Node: Initializing the Minimizer723768 Node: Providing the function to minimize725987 Node: Minimization Iteration726458 Node: Minimization Stopping Parameters728106 Node: Minimization Algorithms729707 Node: Minimization Examples732139 Node: Minimization References and Further Reading735106 Node: Multidimensional Root-Finding735552 Node: Overview of Multidimensional Root Finding737038 Node: Initializing the Multidimensional Solver739218 Node: Providing the multidimensional system of equations to solve742383 Node: Iteration of the multidimensional solver747259 Node: Search Stopping Parameters for the multidimensional solver748862 Node: Algorithms using Derivatives750594 Node: Algorithms without Derivatives755329 Node: Example programs for Multidimensional Root finding758111 Node: References and Further Reading for Multidimensional Root Finding766835 Node: Multidimensional Minimization768064 Node: Multimin Overview769353 Node: Multimin Caveats770931 Node: Initializing the Multidimensional Minimizer771673 Node: Providing a function to minimize773828 Node: Multimin Iteration777046 Node: Multimin Stopping Criteria778576 Node: Multimin Algorithms779767 Node: Multimin Examples782530 Node: Multimin References and Further Reading785625 Node: Least-Squares Fitting786111 Node: Linear regression787042 Node: Linear fitting without a constant term789635 Node: Multi-parameter fitting792027 Node: Fitting Examples795779 Node: Fitting References and Further Reading802553 Node: Nonlinear Least-Squares Fitting803362 Node: Overview of Nonlinear Least-Squares Fitting804813 Node: Initializing the Nonlinear Least-Squares Solver805696 Node: Providing the Function to be Minimized808357 Node: Iteration of the Minimization Algorithm810996 Node: Search Stopping Parameters for Minimization Algorithms812323 Node: Minimization Algorithms using Derivatives814412 Node: Minimization Algorithms without Derivatives817648 Node: Computing the covariance matrix of best fit parameters818033 Node: Example programs for Nonlinear Least-Squares Fitting819091 Node: References and Further Reading for Nonlinear Least-Squares Fitting826030 Node: Physical Constants826753 Node: Fundamental Constants828096 Node: Astronomy and Astrophysics829106 Node: Atomic and Nuclear Physics829755 Node: Measurement of Time831036 Node: Imperial Units831452 Node: Nautical Units831869 Node: Printers Units832184 Node: Volume832469 Node: Mass and Weight832980 Node: Thermal Energy and Power833762 Node: Pressure834169 Node: Viscosity834763 Node: Light and Illumination835025 Node: Radioactivity835598 Node: Physical Constant Examples835928 Node: Physical Constant References and Further Reading837718 Node: IEEE floating-point arithmetic838074 Node: Representation of floating point numbers838657 Node: Setting up your IEEE environment843160 Node: IEEE References and Further Reading849247 Node: Debugging Numerical Programs850245 Node: Using gdb850707 Node: Examining floating point registers854063 Node: Handling floating point exceptions855351 Node: GCC warning options for numerical programs856760 Node: Debugging References860947 Node: Contributors to GSL861489 Node: Autoconf Macros863252 Node: GSL CBLAS Library864051 Node: Level 1 CBLAS Functions864547 Node: Level 2 CBLAS Functions869693 Node: Level 3 CBLAS Functions886055 Node: GSL CBLAS Examples895574 Node: GNU General Public License897170 Node: GNU Free Documentation License916449 Node: Function Index936366 Node: Variable Index1046435 Node: Type Index1047245 Node: Concept Index1049099  End Tag Table