This is gsl-ref.info, produced by makeinfo version 5.1 from gsl-ref.texi. Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 The GSL Team. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections and no cover texts. A copy of the license is included in the section entitled "GNU Free Documentation License". INFO-DIR-SECTION Software libraries START-INFO-DIR-ENTRY * gsl-ref: (gsl-ref). GNU Scientific Library - Reference END-INFO-DIR-ENTRY  Indirect: gsl-ref.info-1: 707 gsl-ref.info-2: 301602 gsl-ref.info-3: 601000 gsl-ref.info-4: 902430 gsl-ref.info-5: 1443412  Tag Table: (Indirect) Node: Top707 Node: Introduction3656 Node: Routines available in GSL4376 Node: GSL is Free Software5989 Node: Obtaining GSL8381 Node: No Warranty9409 Node: Reporting Bugs9915 Ref: Reporting Bugs-Footnote-110867 Node: Further Information10917 Node: Conventions used in this manual11931 Node: Using the library12695 Node: An Example Program13418 Ref: An Example Program-Footnote-114161 Node: Compiling and Linking14268 Node: Linking programs with the library15341 Ref: Linking programs with the library-Footnote-116744 Ref: Linking programs with the library-Footnote-216781 Node: Linking with an alternative BLAS library16834 Node: Shared Libraries17917 Ref: Shared Libraries-Footnote-119392 Node: ANSI C Compliance19440 Node: Inline functions20533 Node: Long double22071 Node: Portability functions23659 Node: Alternative optimized functions25057 Node: Support for different numeric types26515 Node: Compatibility with C++29579 Node: Aliasing of arrays30151 Node: Thread-safety30902 Node: Deprecated Functions31990 Node: Code Reuse32615 Node: Error Handling33260 Node: Error Reporting34060 Node: Error Codes35952 Node: Error Handlers37809 Node: Using GSL error reporting in your own functions41449 Node: Error Reporting Examples43409 Node: Mathematical Functions44621 Node: Mathematical Constants45414 Node: Infinities and Not-a-number46544 Ref: Infinities and Not-a-number-Footnote-147654 Node: Elementary Functions48064 Node: Small integer powers50521 Node: Testing the Sign of Numbers52170 Node: Testing for Odd and Even Numbers52603 Node: Maximum and Minimum functions53162 Node: Approximate Comparison of Floating Point Numbers55443 Node: Complex Numbers56803 Ref: Complex Numbers-Footnote-158148 Node: Representation of complex numbers58212 Node: Properties of complex numbers60152 Node: Complex arithmetic operators61158 Node: Elementary Complex Functions63904 Node: Complex Trigonometric Functions65728 Node: Inverse Complex Trigonometric Functions66949 Node: Complex Hyperbolic Functions69562 Node: Inverse Complex Hyperbolic Functions70858 Node: Complex Number References and Further Reading72846 Node: Polynomials74242 Node: Polynomial Evaluation75046 Node: Divided Difference Representation of Polynomials76457 Node: Quadratic Equations80319 Node: Cubic Equations82256 Node: General Polynomial Equations83839 Node: Roots of Polynomials Examples86196 Node: Roots of Polynomials References and Further Reading87585 Node: Special Functions88631 Node: Special Function Usage90879 Node: The gsl_sf_result struct92063 Node: Special Function Modes93327 Node: Airy Functions and Derivatives94261 Node: Airy Functions94986 Node: Derivatives of Airy Functions96346 Node: Zeros of Airy Functions97858 Node: Zeros of Derivatives of Airy Functions98577 Node: Bessel Functions99336 Node: Regular Cylindrical Bessel Functions100554 Node: Irregular Cylindrical Bessel Functions101878 Node: Regular Modified Cylindrical Bessel Functions103336 Node: Irregular Modified Cylindrical Bessel Functions106164 Node: Regular Spherical Bessel Functions109137 Node: Irregular Spherical Bessel Functions111350 Node: Regular Modified Spherical Bessel Functions113036 Node: Irregular Modified Spherical Bessel Functions115037 Node: Regular Bessel Function - Fractional Order117100 Node: Irregular Bessel Functions - Fractional Order118098 Node: Regular Modified Bessel Functions - Fractional Order118680 Node: Irregular Modified Bessel Functions - Fractional Order119605 Node: Zeros of Regular Bessel Functions120841 Node: Clausen Functions121944 Node: Coulomb Functions122548 Node: Normalized Hydrogenic Bound States123007 Node: Coulomb Wave Functions124120 Node: Coulomb Wave Function Normalization Constant127532 Node: Coupling Coefficients128285 Node: 3-j Symbols129059 Node: 6-j Symbols129663 Node: 9-j Symbols130287 Node: Dawson Function130993 Node: Debye Functions131560 Node: Dilogarithm133334 Node: Real Argument133656 Node: Complex Argument134350 Node: Elementary Operations134820 Node: Elliptic Integrals135644 Node: Definition of Legendre Forms136239 Node: Definition of Carlson Forms137059 Node: Legendre Form of Complete Elliptic Integrals137756 Node: Legendre Form of Incomplete Elliptic Integrals139309 Node: Carlson Forms141496 Node: Elliptic Functions (Jacobi)143043 Node: Error Functions143624 Node: Error Function144100 Node: Complementary Error Function144473 Node: Log Complementary Error Function144933 Node: Probability functions145385 Node: Exponential Functions146638 Node: Exponential Function147033 Node: Relative Exponential Functions148255 Node: Exponentiation With Error Estimate149927 Node: Exponential Integrals151136 Node: Exponential Integral151692 Node: Ei(x)152613 Node: Hyperbolic Integrals153061 Node: Ei_3(x)153716 Node: Trigonometric Integrals154108 Node: Arctangent Integral154707 Node: Fermi-Dirac Function155107 Node: Complete Fermi-Dirac Integrals155481 Node: Incomplete Fermi-Dirac Integrals158046 Node: Gamma and Beta Functions158689 Node: Gamma Functions159324 Node: Factorials162432 Node: Pochhammer Symbol164985 Node: Incomplete Gamma Functions166486 Node: Beta Functions167735 Node: Incomplete Beta Function168487 Node: Gegenbauer Functions169197 Node: Hypergeometric Functions170847 Node: Laguerre Functions175111 Node: Lambert W Functions176674 Node: Legendre Functions and Spherical Harmonics177714 Node: Legendre Polynomials178326 Node: Associated Legendre Polynomials and Spherical Harmonics180382 Node: Conical Functions183211 Node: Radial Functions for Hyperbolic Space185334 Node: Logarithm and Related Functions187511 Node: Mathieu Functions189135 Node: Mathieu Function Workspace190495 Node: Mathieu Function Characteristic Values191308 Node: Angular Mathieu Functions192380 Node: Radial Mathieu Functions193416 Node: Power Function194653 Node: Psi (Digamma) Function195581 Node: Digamma Function196134 Node: Trigamma Function196964 Node: Polygamma Function197541 Node: Synchrotron Functions197925 Node: Transport Functions198688 Node: Trigonometric Functions199835 Node: Circular Trigonometric Functions200481 Node: Trigonometric Functions for Complex Arguments201490 Node: Hyperbolic Trigonometric Functions202526 Node: Conversion Functions203129 Node: Restriction Functions203912 Node: Trigonometric Functions With Error Estimates204843 Node: Zeta Functions205753 Node: Riemann Zeta Function206211 Node: Riemann Zeta Function Minus One206890 Node: Hurwitz Zeta Function207661 Node: Eta Function208166 Node: Special Functions Examples208725 Node: Special Functions References and Further Reading210420 Node: Vectors and Matrices211688 Node: Data types212531 Node: Blocks213734 Node: Block allocation214667 Node: Reading and writing blocks216071 Node: Example programs for blocks218135 Node: Vectors218762 Node: Vector allocation220730 Node: Accessing vector elements222267 Ref: Accessing vector elements-Footnote-1225408 Node: Initializing vector elements225647 Node: Reading and writing vectors226341 Node: Vector views228431 Node: Copying vectors235897 Node: Exchanging elements236757 Node: Vector operations237315 Node: Finding maximum and minimum elements of vectors239088 Node: Vector properties240675 Node: Example programs for vectors241649 Node: Matrices243938 Node: Matrix allocation246878 Node: Accessing matrix elements248464 Node: Initializing matrix elements250413 Node: Reading and writing matrices251210 Node: Matrix views253313 Node: Creating row and column views260449 Node: Copying matrices264926 Node: Copying rows and columns265522 Node: Exchanging rows and columns267199 Node: Matrix operations268677 Node: Finding maximum and minimum elements of matrices270647 Node: Matrix properties272547 Node: Example programs for matrices273650 Node: Vector and Matrix References and Further Reading277721 Node: Permutations278207 Node: The Permutation struct279544 Node: Permutation allocation280047 Node: Accessing permutation elements281483 Node: Permutation properties282326 Node: Permutation functions283031 Node: Applying Permutations284288 Node: Reading and writing permutations286215 Ref: Reading and writing permutations-Footnote-1288431 Node: Permutations in cyclic form288543 Node: Permutation Examples291970 Node: Permutation References and Further Reading294513 Node: Combinations295187 Node: The Combination struct296072 Node: Combination allocation296623 Node: Accessing combination elements298410 Node: Combination properties299069 Node: Combination functions299955 Node: Reading and writing combinations301602 Ref: Reading and writing combinations-Footnote-1303835 Node: Combination Examples303947 Node: Combination References and Further Reading305255 Node: Multisets305658 Node: The Multiset struct306584 Node: Multiset allocation307105 Node: Accessing multiset elements308800 Node: Multiset properties309429 Node: Multiset functions310242 Node: Reading and writing multisets311152 Ref: Reading and writing multisets-Footnote-1313303 Node: Multiset Examples313415 Node: Sorting315523 Node: Sorting objects316522 Node: Sorting vectors319435 Node: Selecting the k smallest or largest elements322596 Node: Computing the rank325937 Node: Sorting Examples327138 Node: Sorting References and Further Reading328782 Node: BLAS Support329328 Ref: BLAS Support-Footnote-1332714 Node: GSL BLAS Interface332883 Node: Level 1 GSL BLAS Interface333376 Node: Level 2 GSL BLAS Interface340116 Node: Level 3 GSL BLAS Interface349406 Node: BLAS Examples360504 Node: BLAS References and Further Reading361925 Node: Linear Algebra363267 Node: LU Decomposition364461 Node: QR Decomposition369697 Node: QR Decomposition with Column Pivoting376006 Node: Singular Value Decomposition380470 Node: Cholesky Decomposition384890 Node: Tridiagonal Decomposition of Real Symmetric Matrices388132 Node: Tridiagonal Decomposition of Hermitian Matrices389980 Node: Hessenberg Decomposition of Real Matrices391932 Node: Hessenberg-Triangular Decomposition of Real Matrices394344 Node: Bidiagonalization395467 Node: Householder Transformations397892 Node: Householder solver for linear systems400244 Node: Tridiagonal Systems401151 Node: Balancing404024 Node: Linear Algebra Examples404827 Node: Linear Algebra References and Further Reading406816 Node: Eigensystems408463 Node: Real Symmetric Matrices409692 Node: Complex Hermitian Matrices412206 Node: Real Nonsymmetric Matrices414835 Node: Real Generalized Symmetric-Definite Eigensystems420570 Node: Complex Generalized Hermitian-Definite Eigensystems423449 Node: Real Generalized Nonsymmetric Eigensystems426163 Node: Sorting Eigenvalues and Eigenvectors432770 Node: Eigenvalue and Eigenvector Examples436048 Node: Eigenvalue and Eigenvector References441439 Node: Fast Fourier Transforms442466 Node: Mathematical Definitions443692 Node: Overview of complex data FFTs446097 Node: Radix-2 FFT routines for complex data449042 Node: Mixed-radix FFT routines for complex data453261 Node: Overview of real data FFTs462594 Node: Radix-2 FFT routines for real data465029 Node: Mixed-radix FFT routines for real data469453 Node: FFT References and Further Reading480138 Node: Numerical Integration483008 Node: Numerical Integration Introduction484806 Node: Integrands without weight functions487081 Node: Integrands with weight functions487911 Node: Integrands with singular weight functions488622 Node: QNG non-adaptive Gauss-Kronrod integration489540 Node: QAG adaptive integration490822 Node: QAGS adaptive integration with singularities493611 Node: QAGP adaptive integration with known singular points495411 Node: QAGI adaptive integration on infinite intervals496725 Node: QAWC adaptive integration for Cauchy principal values499005 Node: QAWS adaptive integration for singular functions500175 Node: QAWO adaptive integration for oscillatory functions503401 Node: QAWF adaptive integration for Fourier integrals507189 Node: CQUAD doubly-adaptive integration509813 Node: Fixed order Gauss-Legendre integration513071 Node: Numerical integration error codes515059 Node: Numerical integration examples515815 Node: Numerical integration References and Further Reading517861 Node: Random Number Generation518852 Node: General comments on random numbers520386 Node: The Random Number Generator Interface522338 Node: Random number generator initialization523696 Node: Sampling from a random number generator526067 Node: Auxiliary random number generator functions529311 Node: Random number environment variables531631 Node: Copying random number generator state534156 Node: Reading and writing random number generator state535130 Node: Random number generator algorithms536524 Node: Unix random number generators546466 Node: Other random number generators550180 Node: Random Number Generator Performance558831 Node: Random Number Generator Examples559952 Node: Random Number References and Further Reading561508 Node: Random Number Acknowledgements562803 Node: Quasi-Random Sequences563289 Node: Quasi-random number generator initialization564409 Node: Sampling from a quasi-random number generator565424 Node: Auxiliary quasi-random number generator functions566121 Node: Saving and resorting quasi-random number generator state567067 Node: Quasi-random number generator algorithms567875 Node: Quasi-random number generator examples569013 Node: Quasi-random number references569998 Node: Random Number Distributions570524 Node: Random Number Distribution Introduction573855 Node: The Gaussian Distribution575650 Node: The Gaussian Tail Distribution578310 Node: The Bivariate Gaussian Distribution579974 Node: The Exponential Distribution581288 Node: The Laplace Distribution582427 Node: The Exponential Power Distribution583519 Node: The Cauchy Distribution584787 Node: The Rayleigh Distribution586036 Node: The Rayleigh Tail Distribution587208 Node: The Landau Distribution588081 Node: The Levy alpha-Stable Distributions589036 Node: The Levy skew alpha-Stable Distribution590094 Node: The Gamma Distribution591706 Node: The Flat (Uniform) Distribution593361 Node: The Lognormal Distribution594509 Node: The Chi-squared Distribution595857 Node: The F-distribution597262 Node: The t-distribution598899 Node: The Beta Distribution601000 Node: The Logistic Distribution602159 Node: The Pareto Distribution603288 Node: Spherical Vector Distributions604458 Node: The Weibull Distribution607294 Node: The Type-1 Gumbel Distribution608488 Node: The Type-2 Gumbel Distribution609724 Node: The Dirichlet Distribution610955 Node: General Discrete Distributions612623 Node: The Poisson Distribution616491 Node: The Bernoulli Distribution617498 Node: The Binomial Distribution618249 Node: The Multinomial Distribution619452 Node: The Negative Binomial Distribution621226 Node: The Pascal Distribution622591 Node: The Geometric Distribution623749 Node: The Hypergeometric Distribution624997 Node: The Logarithmic Distribution626650 Node: Shuffling and Sampling627442 Node: Random Number Distribution Examples630254 Node: Random Number Distribution References and Further Reading633450 Node: Statistics635594 Node: Mean and standard deviation and variance637080 Node: Absolute deviation640588 Node: Higher moments (skewness and kurtosis)641880 Node: Autocorrelation644013 Node: Covariance644826 Node: Correlation645792 Node: Weighted Samples647161 Node: Maximum and Minimum values653011 Node: Median and Percentiles655750 Node: Example statistical programs658165 Node: Statistics References and Further Reading660825 Node: Histograms662033 Node: The histogram struct663886 Node: Histogram allocation665690 Node: Copying Histograms668647 Node: Updating and accessing histogram elements669323 Node: Searching histogram ranges672592 Node: Histogram Statistics673592 Node: Histogram Operations675458 Node: Reading and writing histograms677533 Node: Resampling from histograms680575 Node: The histogram probability distribution struct681374 Node: Example programs for histograms684408 Node: Two dimensional histograms686471 Node: The 2D histogram struct687192 Node: 2D Histogram allocation689002 Node: Copying 2D Histograms691076 Node: Updating and accessing 2D histogram elements691781 Node: Searching 2D histogram ranges695438 Node: 2D Histogram Statistics696452 Node: 2D Histogram Operations699317 Node: Reading and writing 2D histograms701495 Node: Resampling from 2D histograms705124 Node: Example programs for 2D histograms708145 Node: N-tuples709972 Node: The ntuple struct711295 Node: Creating ntuples711774 Node: Opening an existing ntuple file712441 Node: Writing ntuples713070 Node: Reading ntuples713531 Node: Closing an ntuple file713862 Node: Histogramming ntuple values714202 Node: Example ntuple programs716212 Node: Ntuple References and Further Reading719541 Node: Monte Carlo Integration719862 Node: Monte Carlo Interface721173 Node: PLAIN Monte Carlo723800 Node: MISER726255 Ref: MISER-Footnote-1732935 Node: VEGAS733060 Node: Monte Carlo Examples742621 Node: Monte Carlo Integration References and Further Reading748608 Node: Simulated Annealing749390 Node: Simulated Annealing algorithm750606 Node: Simulated Annealing functions751758 Node: Examples with Simulated Annealing756288 Node: Trivial example756857 Node: Traveling Salesman Problem759508 Node: Simulated Annealing References and Further Reading762821 Node: Ordinary Differential Equations763232 Node: Defining the ODE System764472 Node: Stepping Functions767300 Node: Adaptive Step-size Control774270 Node: Evolution780347 Node: Driver784476 Node: ODE Example programs788710 Node: ODE References and Further Reading793756 Node: Interpolation795520 Node: Introduction to Interpolation796721 Node: Interpolation Functions797162 Node: Interpolation Types798369 Node: Index Look-up and Acceleration801091 Node: Evaluation of Interpolating Functions803143 Node: Higher-level Interface805639 Node: Interpolation Example programs807679 Node: Interpolation References and Further Reading810899 Node: Numerical Differentiation811471 Node: Numerical Differentiation functions812066 Node: Numerical Differentiation Examples814923 Node: Numerical Differentiation References816336 Node: Chebyshev Approximations816887 Node: Chebyshev Definitions817968 Node: Creation and Calculation of Chebyshev Series818757 Node: Auxiliary Functions for Chebyshev Series819742 Node: Chebyshev Series Evaluation820482 Node: Derivatives and Integrals821862 Node: Chebyshev Approximation Examples823100 Node: Chebyshev Approximation References and Further Reading824596 Node: Series Acceleration825046 Node: Acceleration functions825823 Node: Acceleration functions without error estimation828150 Node: Example of accelerating a series830774 Node: Series Acceleration References833120 Node: Wavelet Transforms834008 Node: DWT Definitions834607 Node: DWT Initialization835559 Node: DWT Transform Functions838207 Node: DWT in one dimension838758 Node: DWT in two dimension840779 Node: DWT Examples845358 Node: DWT References847175 Node: Discrete Hankel Transforms849339 Node: Discrete Hankel Transform Definition849813 Node: Discrete Hankel Transform Functions852016 Node: Discrete Hankel Transform References853681 Node: One dimensional Root-Finding854085 Node: Root Finding Overview855391 Node: Root Finding Caveats857247 Node: Initializing the Solver859016 Node: Providing the function to solve861649 Node: Search Bounds and Guesses865203 Node: Root Finding Iteration866066 Node: Search Stopping Parameters867917 Node: Root Bracketing Algorithms870428 Node: Root Finding Algorithms using Derivatives873726 Ref: Root Finding Algorithms using Derivatives-Footnote-1877269 Node: Root Finding Examples877425 Node: Root Finding References and Further Reading884713 Node: One dimensional Minimization885352 Node: Minimization Overview886695 Node: Minimization Caveats888402 Node: Initializing the Minimizer889740 Node: Providing the function to minimize891981 Node: Minimization Iteration892459 Node: Minimization Stopping Parameters894597 Node: Minimization Algorithms896202 Node: Minimization Examples898796 Node: Minimization References and Further Reading902430 Node: Multidimensional Root-Finding902886 Node: Overview of Multidimensional Root Finding904394 Node: Initializing the Multidimensional Solver906581 Node: Providing the multidimensional system of equations to solve909811 Node: Iteration of the multidimensional solver914723 Node: Search Stopping Parameters for the multidimensional solver917004 Node: Algorithms using Derivatives918742 Node: Algorithms without Derivatives923567 Node: Example programs for Multidimensional Root finding926657 Node: References and Further Reading for Multidimensional Root Finding935281 Node: Multidimensional Minimization936520 Node: Multimin Overview937930 Node: Multimin Caveats940010 Node: Initializing the Multidimensional Minimizer940759 Node: Providing a function to minimize943993 Node: Multimin Iteration948054 Node: Multimin Stopping Criteria950248 Node: Multimin Algorithms with Derivatives951819 Node: Multimin Algorithms without Derivatives955350 Node: Multimin Examples958352 Node: Multimin References and Further Reading964922 Node: Least-Squares Fitting965786 Node: Fitting Overview966872 Node: Linear regression969205 Node: Linear fitting without a constant term971807 Node: Multi-parameter fitting974001 Node: Robust linear regression979860 Node: Troubleshooting990108 Node: Fitting Examples990731 Node: Fitting References and Further Reading1001255 Node: Nonlinear Least-Squares Fitting1002627 Node: Overview of Nonlinear Least-Squares Fitting1004151 Node: Initializing the Nonlinear Least-Squares Solver1005618 Node: Providing the Function to be Minimized1008483 Node: Finite Difference Jacobian1011946 Node: Iteration of the Minimization Algorithm1013817 Node: Search Stopping Parameters for Minimization Algorithms1015657 Node: High Level Driver1017730 Node: Minimization Algorithms using Derivatives1018636 Ref: Minimization Algorithms using Derivatives-Footnote-11022084 Node: Minimization Algorithms without Derivatives1022173 Node: Computing the covariance matrix of best fit parameters1022565 Node: Example programs for Nonlinear Least-Squares Fitting1024602 Node: References and Further Reading for Nonlinear Least-Squares Fitting1032335 Node: Basis Splines1033071 Node: Overview of B-splines1033950 Node: Initializing the B-splines solver1035266 Node: Constructing the knots vector1036647 Node: Evaluation of B-spline basis functions1037435 Node: Evaluation of B-spline basis function derivatives1039083 Node: Working with the Greville abscissae1041053 Node: Example programs for B-splines1042047 Node: References and Further Reading1046042 Node: Physical Constants1046848 Node: Fundamental Constants1048427 Node: Astronomy and Astrophysics1049567 Node: Atomic and Nuclear Physics1050232 Node: Measurement of Time1051883 Node: Imperial Units1052313 Node: Speed and Nautical Units1052755 Node: Printers Units1053259 Node: Volume Area and Length1053582 Node: Mass and Weight1054268 Node: Thermal Energy and Power1055087 Node: Pressure1055510 Node: Viscosity1056123 Node: Light and Illumination1056399 Node: Radioactivity1056991 Node: Force and Energy1057326 Node: Prefixes1057730 Node: Physical Constant Examples1058473 Node: Physical Constant References and Further Reading1060262 Node: IEEE floating-point arithmetic1060992 Node: Representation of floating point numbers1061584 Node: Setting up your IEEE environment1066070 Node: IEEE References and Further Reading1073043 Node: Debugging Numerical Programs1074203 Node: Using gdb1074720 Node: Examining floating point registers1078063 Node: Handling floating point exceptions1079348 Node: GCC warning options for numerical programs1080760 Node: Debugging References1084708 Node: Contributors to GSL1085420 Node: Autoconf Macros1089914 Node: GSL CBLAS Library1093946 Node: Level 1 CBLAS Functions1094486 Node: Level 2 CBLAS Functions1099788 Node: Level 3 CBLAS Functions1116418 Node: GSL CBLAS Examples1126080 Node: GNU General Public License1127643 Node: GNU Free Documentation License1165217 Node: Function Index1190366 Node: Variable Index1443412 Node: Type Index1446715 Node: Concept Index1461784  End Tag Table