This is gsl-ref.info, produced by makeinfo version 4.11 from gsl-ref.texi. INFO-DIR-SECTION Software libraries START-INFO-DIR-ENTRY * gsl-ref: (gsl-ref). GNU Scientific Library - Reference END-INFO-DIR-ENTRY Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 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 the Invariant Sections being "GNU General Public License" and "Free Software Needs Free Documentation", the Front-Cover text being "A GNU Manual", and with the Back-Cover Text being (a) (see below). A copy of the license is included in the section entitled "GNU Free Documentation License". (a) The Back-Cover Text is: "You have the freedom to copy and modify this GNU Manual."  Indirect: gsl-ref.info-1: 936 gsl-ref.info-2: 300349 gsl-ref.info-3: 599961 gsl-ref.info-4: 899144 gsl-ref.info-5: 1139739 gsl-ref.info-6: 1401858  Tag Table: (Indirect) Node: Top936 Node: Introduction3641 Node: Routines available in GSL4293 Node: GSL is Free Software6208 Node: Obtaining GSL8600 Node: No Warranty9627 Node: Reporting Bugs10133 Ref: Reporting Bugs-Footnote-111089 Node: Further Information11139 Node: Conventions used in this manual12152 Node: Using the library12916 Node: An Example Program13502 Ref: An Example Program-Footnote-114245 Node: Compiling and Linking14352 Node: Linking programs with the library15422 Node: Linking with an alternative BLAS library16567 Node: Shared Libraries17649 Ref: Shared Libraries-Footnote-119124 Node: ANSI C Compliance19172 Node: Inline functions20265 Node: Long double21803 Node: Portability functions23391 Node: Alternative optimized functions24788 Node: Support for different numeric types26246 Node: Compatibility with C++29309 Node: Aliasing of arrays29881 Node: Thread-safety30632 Node: Deprecated Functions31719 Node: Code Reuse32344 Node: Error Handling32988 Node: Error Reporting33738 Node: Error Codes35630 Node: Error Handlers37486 Node: Using GSL error reporting in your own functions41123 Node: Error Reporting Examples43081 Node: Mathematical Functions44293 Node: Mathematical Constants45054 Node: Infinities and Not-a-number46184 Ref: Infinities and Not-a-number-Footnote-147291 Node: Elementary Functions47701 Node: Small integer powers50148 Node: Testing the Sign of Numbers51732 Node: Testing for Odd and Even Numbers52163 Node: Maximum and Minimum functions52719 Node: Approximate Comparison of Floating Point Numbers54994 Node: Complex Numbers56353 Ref: Complex Numbers-Footnote-157679 Node: Representation of complex numbers57743 Node: Properties of complex numbers59680 Node: Complex arithmetic operators60685 Node: Elementary Complex Functions63431 Node: Complex Trigonometric Functions65251 Node: Inverse Complex Trigonometric Functions66472 Node: Complex Hyperbolic Functions69078 Node: Inverse Complex Hyperbolic Functions70374 Node: Complex Number References and Further Reading72362 Node: Polynomials73758 Node: Polynomial Evaluation74525 Node: Divided Difference Representation of Polynomials75935 Node: Quadratic Equations77684 Node: Cubic Equations79624 Node: General Polynomial Equations81206 Node: Roots of Polynomials Examples83551 Node: Roots of Polynomials References and Further Reading84940 Node: Special Functions85885 Node: Special Function Usage87879 Node: The gsl_sf_result struct89062 Node: Special Function Modes90326 Node: Airy Functions and Derivatives91262 Node: Airy Functions91958 Node: Derivatives of Airy Functions93318 Node: Zeros of Airy Functions94830 Node: Zeros of Derivatives of Airy Functions95549 Node: Bessel Functions96308 Node: Regular Cylindrical Bessel Functions97500 Node: Irregular Cylindrical Bessel Functions98824 Node: Regular Modified Cylindrical Bessel Functions100282 Node: Irregular Modified Cylindrical Bessel Functions103109 Node: Regular Spherical Bessel Functions106077 Node: Irregular Spherical Bessel Functions108291 Node: Regular Modified Spherical Bessel Functions109978 Node: Irregular Modified Spherical Bessel Functions111981 Node: Regular Bessel Function - Fractional Order114044 Node: Irregular Bessel Functions - Fractional Order115042 Node: Regular Modified Bessel Functions - Fractional Order115624 Node: Irregular Modified Bessel Functions - Fractional Order116549 Node: Zeros of Regular Bessel Functions117785 Node: Clausen Functions118888 Node: Coulomb Functions119492 Node: Normalized Hydrogenic Bound States119941 Node: Coulomb Wave Functions121054 Node: Coulomb Wave Function Normalization Constant124464 Node: Coupling Coefficients125217 Node: 3-j Symbols125940 Node: 6-j Symbols126544 Node: 9-j Symbols127168 Node: Dawson Function127874 Node: Debye Functions128441 Node: Dilogarithm130215 Node: Real Argument130510 Node: Complex Argument131203 Node: Elementary Operations131672 Node: Elliptic Integrals132496 Node: Definition of Legendre Forms133068 Node: Definition of Carlson Forms133888 Node: Legendre Form of Complete Elliptic Integrals134585 Node: Legendre Form of Incomplete Elliptic Integrals136153 Node: Carlson Forms138341 Node: Elliptic Functions (Jacobi)139888 Node: Error Functions140469 Node: Error Function140920 Node: Complementary Error Function141293 Node: Log Complementary Error Function141753 Node: Probability functions142205 Node: Exponential Functions143458 Node: Exponential Function143841 Node: Relative Exponential Functions145064 Node: Exponentiation With Error Estimate146736 Node: Exponential Integrals147945 Node: Exponential Integral148427 Node: Ei(x)149354 Node: Hyperbolic Integrals149802 Node: Ei_3(x)150458 Node: Trigonometric Integrals150850 Node: Arctangent Integral151449 Node: Fermi-Dirac Function151849 Node: Complete Fermi-Dirac Integrals152219 Node: Incomplete Fermi-Dirac Integrals154787 Node: Gamma and Beta Functions155430 Node: Gamma Functions156003 Node: Factorials159112 Node: Pochhammer Symbol161665 Node: Incomplete Gamma Functions163171 Node: Beta Functions164420 Node: Incomplete Beta Function165172 Node: Gegenbauer Functions165883 Node: Hypergeometric Functions167533 Node: Laguerre Functions171796 Node: Lambert W Functions173359 Node: Legendre Functions and Spherical Harmonics174397 Node: Legendre Polynomials174986 Node: Associated Legendre Polynomials and Spherical Harmonics177041 Node: Conical Functions179878 Node: Radial Functions for Hyperbolic Space182001 Node: Logarithm and Related Functions184178 Node: Mathieu Functions185802 Node: Mathieu Function Workspace187151 Node: Mathieu Function Characteristic Values187965 Node: Angular Mathieu Functions189037 Node: Radial Mathieu Functions190073 Node: Power Function191315 Node: Psi (Digamma) Function192242 Node: Digamma Function192762 Node: Trigamma Function193592 Node: Polygamma Function194169 Node: Synchrotron Functions194553 Node: Transport Functions195316 Node: Trigonometric Functions196463 Node: Circular Trigonometric Functions197086 Node: Trigonometric Functions for Complex Arguments198095 Node: Hyperbolic Trigonometric Functions199131 Node: Conversion Functions199734 Node: Restriction Functions200517 Node: Trigonometric Functions With Error Estimates201449 Node: Zeta Functions202359 Node: Riemann Zeta Function202817 Node: Riemann Zeta Function Minus One203496 Node: Hurwitz Zeta Function204267 Node: Eta Function204772 Node: Special Functions Examples205331 Node: Special Functions References and Further Reading207025 Node: Vectors and Matrices208295 Node: Data types209054 Node: Blocks210257 Node: Block allocation211173 Node: Reading and writing blocks212657 Node: Example programs for blocks214721 Node: Vectors215348 Node: Vector allocation217234 Node: Accessing vector elements218852 Ref: Accessing vector elements-Footnote-1221994 Node: Initializing vector elements222233 Node: Reading and writing vectors222927 Node: Vector views225017 Node: Copying vectors232459 Node: Exchanging elements233319 Node: Vector operations233877 Node: Finding maximum and minimum elements of vectors235330 Node: Vector properties236916 Node: Example programs for vectors237681 Node: Matrices239970 Node: Matrix allocation242828 Node: Accessing matrix elements244495 Node: Initializing matrix elements246444 Node: Reading and writing matrices247240 Node: Matrix views249343 Node: Creating row and column views256483 Node: Copying matrices260954 Node: Copying rows and columns261550 Node: Exchanging rows and columns263227 Node: Matrix operations264705 Node: Finding maximum and minimum elements of matrices266345 Node: Matrix properties268249 Node: Example programs for matrices269141 Node: Vector and Matrix References and Further Reading273212 Node: Permutations273698 Node: The Permutation struct274986 Node: Permutation allocation275490 Node: Accessing permutation elements276924 Node: Permutation properties277767 Node: Permutation functions278472 Node: Applying Permutations279729 Node: Reading and writing permutations281655 Ref: Reading and writing permutations-Footnote-1283870 Node: Permutations in cyclic form283982 Node: Permutation Examples287407 Node: Permutation References and Further Reading289950 Node: Combinations290624 Node: The Combination struct291469 Node: Combination allocation292021 Node: Accessing combination elements293806 Node: Combination properties294465 Node: Combination functions295351 Node: Reading and writing combinations296291 Ref: Reading and writing combinations-Footnote-1298525 Node: Combination Examples298637 Node: Combination References and Further Reading299945 Node: Sorting300349 Node: Sorting objects301299 Node: Sorting vectors304210 Node: Selecting the k smallest or largest elements306786 Node: Computing the rank310120 Node: Sorting Examples311321 Node: Sorting References and Further Reading312965 Node: BLAS Support313511 Ref: BLAS Support-Footnote-1316890 Node: GSL BLAS Interface317058 Node: Level 1 GSL BLAS Interface317545 Node: Level 2 GSL BLAS Interface324284 Node: Level 3 GSL BLAS Interface333578 Node: BLAS Examples344718 Node: BLAS References and Further Reading346139 Node: Linear Algebra347486 Node: LU Decomposition348611 Node: QR Decomposition353856 Node: QR Decomposition with Column Pivoting360165 Node: Singular Value Decomposition364620 Node: Cholesky Decomposition368616 Node: Tridiagonal Decomposition of Real Symmetric Matrices371701 Node: Tridiagonal Decomposition of Hermitian Matrices373549 Node: Hessenberg Decomposition of Real Matrices375501 Node: Hessenberg-Triangular Decomposition of Real Matrices377911 Node: Bidiagonalization379031 Node: Householder Transformations381456 Node: Householder solver for linear systems383808 Node: Tridiagonal Systems384713 Node: Balancing387585 Node: Linear Algebra Examples388386 Node: Linear Algebra References and Further Reading390374 Node: Eigensystems392019 Node: Real Symmetric Matrices393233 Node: Complex Hermitian Matrices395747 Node: Real Nonsymmetric Matrices398374 Node: Real Generalized Symmetric-Definite Eigensystems403587 Node: Complex Generalized Hermitian-Definite Eigensystems406459 Node: Real Generalized Nonsymmetric Eigensystems409166 Node: Sorting Eigenvalues and Eigenvectors415750 Node: Eigenvalue and Eigenvector Examples419027 Node: Eigenvalue and Eigenvector References424417 Node: Fast Fourier Transforms425442 Node: Mathematical Definitions426651 Node: Overview of complex data FFTs429056 Node: Radix-2 FFT routines for complex data431999 Node: Mixed-radix FFT routines for complex data436211 Node: Overview of real data FFTs445540 Node: Radix-2 FFT routines for real data447974 Node: Mixed-radix FFT routines for real data452403 Node: FFT References and Further Reading463077 Node: Numerical Integration465946 Node: Numerical Integration Introduction467491 Node: Integrands without weight functions469760 Node: Integrands with weight functions470589 Node: Integrands with singular weight functions471300 Node: QNG non-adaptive Gauss-Kronrod integration472218 Node: QAG adaptive integration473499 Node: QAGS adaptive integration with singularities476130 Node: QAGP adaptive integration with known singular points477930 Node: QAGI adaptive integration on infinite intervals479244 Node: QAWC adaptive integration for Cauchy principal values481537 Node: QAWS adaptive integration for singular functions482712 Node: QAWO adaptive integration for oscillatory functions485934 Node: QAWF adaptive integration for Fourier integrals489713 Node: Numerical integration error codes492338 Node: Numerical integration examples493106 Node: Numerical integration References and Further Reading495152 Node: Random Number Generation495881 Node: General comments on random numbers497383 Node: The Random Number Generator Interface499333 Node: Random number generator initialization500691 Node: Sampling from a random number generator503064 Node: Auxiliary random number generator functions506305 Node: Random number environment variables508625 Node: Copying random number generator state511151 Node: Reading and writing random number generator state512126 Node: Random number generator algorithms513520 Node: Unix random number generators523329 Node: Other random number generators527047 Node: Random Number Generator Performance535680 Node: Random Number Generator Examples536998 Node: Random Number References and Further Reading538554 Node: Random Number Acknowledgements539851 Node: Quasi-Random Sequences540337 Node: Quasi-random number generator initialization541444 Node: Sampling from a quasi-random number generator542459 Node: Auxiliary quasi-random number generator functions543156 Node: Saving and resorting quasi-random number generator state544102 Node: Quasi-random number generator algorithms544910 Node: Quasi-random number generator examples546046 Node: Quasi-random number references547031 Node: Random Number Distributions547557 Node: Random Number Distribution Introduction550756 Node: The Gaussian Distribution552551 Node: The Gaussian Tail Distribution555212 Node: The Bivariate Gaussian Distribution556881 Node: The Exponential Distribution558204 Node: The Laplace Distribution559343 Node: The Exponential Power Distribution560440 Node: The Cauchy Distribution561708 Node: The Rayleigh Distribution562957 Node: The Rayleigh Tail Distribution564129 Node: The Landau Distribution565001 Node: The Levy alpha-Stable Distributions565956 Node: The Levy skew alpha-Stable Distribution567014 Node: The Gamma Distribution568626 Node: The Flat (Uniform) Distribution570283 Node: The Lognormal Distribution571435 Node: The Chi-squared Distribution572783 Node: The F-distribution574187 Node: The t-distribution575823 Node: The Beta Distribution577217 Node: The Logistic Distribution578376 Node: The Pareto Distribution579505 Node: Spherical Vector Distributions580675 Node: The Weibull Distribution583509 Node: The Type-1 Gumbel Distribution584703 Node: The Type-2 Gumbel Distribution585940 Node: The Dirichlet Distribution587171 Node: General Discrete Distributions588838 Node: The Poisson Distribution592705 Node: The Bernoulli Distribution593713 Node: The Binomial Distribution594464 Node: The Multinomial Distribution595669 Node: The Negative Binomial Distribution597438 Node: The Pascal Distribution598803 Node: The Geometric Distribution599961 Node: The Hypergeometric Distribution601209 Node: The Logarithmic Distribution602862 Node: Shuffling and Sampling603653 Node: Random Number Distribution Examples606472 Node: Random Number Distribution References and Further Reading609668 Node: Statistics611811 Node: Mean and standard deviation and variance613211 Node: Absolute deviation616719 Node: Higher moments (skewness and kurtosis)618011 Node: Autocorrelation620144 Node: Covariance620958 Node: Correlation621934 Node: Weighted Samples622643 Node: Maximum and Minimum values628493 Node: Median and Percentiles631232 Node: Example statistical programs633647 Node: Statistics References and Further Reading636308 Node: Histograms637516 Node: The histogram struct639276 Node: Histogram allocation641079 Node: Copying Histograms644040 Node: Updating and accessing histogram elements644716 Node: Searching histogram ranges647983 Node: Histogram Statistics648982 Node: Histogram Operations650842 Node: Reading and writing histograms652914 Node: Resampling from histograms655956 Node: The histogram probability distribution struct656754 Node: Example programs for histograms659785 Node: Two dimensional histograms661848 Node: The 2D histogram struct662569 Node: 2D Histogram allocation664377 Node: Copying 2D Histograms666448 Node: Updating and accessing 2D histogram elements667153 Node: Searching 2D histogram ranges670809 Node: 2D Histogram Statistics671828 Node: 2D Histogram Operations674687 Node: Reading and writing 2D histograms676861 Node: Resampling from 2D histograms680490 Node: Example programs for 2D histograms683509 Node: N-tuples685337 Node: The ntuple struct686593 Node: Creating ntuples687071 Node: Opening an existing ntuple file687738 Node: Writing ntuples688366 Node: Reading ntuples688827 Node: Closing an ntuple file689158 Node: Histogramming ntuple values689498 Node: Example ntuple programs691506 Node: Ntuple References and Further Reading694835 Node: Monte Carlo Integration695156 Node: Monte Carlo Interface696401 Node: PLAIN Monte Carlo699024 Node: MISER701474 Ref: MISER-Footnote-1708150 Node: VEGAS708275 Node: Monte Carlo Examples717826 Node: Monte Carlo Integration References and Further Reading723811 Node: Simulated Annealing724593 Node: Simulated Annealing algorithm725800 Node: Simulated Annealing functions726954 Node: Examples with Simulated Annealing731489 Node: Trivial example732043 Node: Traveling Salesman Problem734694 Node: Simulated Annealing References and Further Reading738007 Node: Ordinary Differential Equations738418 Node: Defining the ODE System739239 Node: Stepping Functions741431 Node: Adaptive Step-size Control745776 Node: Evolution751234 Node: ODE Example programs754851 Node: ODE References and Further Reading760090 Node: Interpolation760729 Node: Introduction to Interpolation761900 Node: Interpolation Functions762341 Node: Interpolation Types763548 Node: Index Look-up and Acceleration766157 Node: Evaluation of Interpolating Functions768209 Node: Higher-level Interface770593 Node: Interpolation Example programs772632 Node: Interpolation References and Further Reading775848 Node: Numerical Differentiation776421 Node: Numerical Differentiation functions777010 Node: Numerical Differentiation Examples779868 Node: Numerical Differentiation References781281 Node: Chebyshev Approximations781832 Node: Chebyshev Definitions782893 Node: Creation and Calculation of Chebyshev Series783683 Node: Auxiliary Functions for Chebyshev Series784668 Node: Chebyshev Series Evaluation785408 Node: Derivatives and Integrals786788 Node: Chebyshev Approximation Examples788030 Node: Chebyshev Approximation References and Further Reading789526 Node: Series Acceleration789975 Node: Acceleration functions790740 Node: Acceleration functions without error estimation793065 Node: Example of accelerating a series795675 Node: Series Acceleration References798021 Node: Wavelet Transforms798909 Node: DWT Definitions799450 Node: DWT Initialization800402 Node: DWT Transform Functions802981 Node: DWT in one dimension803516 Node: DWT in two dimension805535 Node: DWT Examples810113 Node: DWT References811930 Node: Discrete Hankel Transforms814093 Node: Discrete Hankel Transform Definition814561 Node: Discrete Hankel Transform Functions816700 Node: Discrete Hankel Transform References818237 Node: One dimensional Root-Finding818641 Node: Root Finding Overview819901 Node: Root Finding Caveats821759 Node: Initializing the Solver823526 Node: Providing the function to solve826159 Node: Search Bounds and Guesses829925 Node: Root Finding Iteration830788 Node: Search Stopping Parameters832637 Node: Root Bracketing Algorithms835151 Node: Root Finding Algorithms using Derivatives838449 Ref: Root Finding Algorithms using Derivatives-Footnote-1841991 Node: Root Finding Examples842146 Node: Root Finding References and Further Reading849434 Node: One dimensional Minimization850073 Node: Minimization Overview851375 Node: Minimization Caveats853081 Node: Initializing the Minimizer854418 Node: Providing the function to minimize856659 Node: Minimization Iteration857137 Node: Minimization Stopping Parameters859273 Node: Minimization Algorithms860882 Node: Minimization Examples863486 Node: Minimization References and Further Reading866413 Node: Multidimensional Root-Finding866869 Node: Overview of Multidimensional Root Finding868358 Node: Initializing the Multidimensional Solver870545 Node: Providing the multidimensional system of equations to solve873776 Node: Iteration of the multidimensional solver878681 Node: Search Stopping Parameters for the multidimensional solver880960 Node: Algorithms using Derivatives882697 Node: Algorithms without Derivatives887443 Node: Example programs for Multidimensional Root finding890531 Node: References and Further Reading for Multidimensional Root Finding899144 Node: Multidimensional Minimization900383 Node: Multimin Overview901736 Node: Multimin Caveats903813 Node: Initializing the Multidimensional Minimizer904562 Node: Providing a function to minimize907723 Node: Multimin Iteration911775 Node: Multimin Stopping Criteria913766 Node: Multimin Algorithms with Derivatives915335 Node: Multimin Algorithms without Derivatives918863 Node: Multimin Examples921788 Node: Multimin References and Further Reading928354 Node: Least-Squares Fitting929219 Node: Fitting Overview930218 Node: Linear regression932552 Node: Linear fitting without a constant term935154 Node: Multi-parameter fitting937347 Node: Fitting Examples942997 Node: Fitting References and Further Reading949822 Node: Nonlinear Least-Squares Fitting950644 Node: Overview of Nonlinear Least-Squares Fitting952094 Node: Initializing the Nonlinear Least-Squares Solver953561 Node: Providing the Function to be Minimized956426 Node: Iteration of the Minimization Algorithm959363 Node: Search Stopping Parameters for Minimization Algorithms961213 Node: Minimization Algorithms using Derivatives963308 Node: Minimization Algorithms without Derivatives966751 Node: Computing the covariance matrix of best fit parameters967143 Node: Example programs for Nonlinear Least-Squares Fitting969177 Node: References and Further Reading for Nonlinear Least-Squares Fitting976905 Node: Basis Splines977641 Node: Overview of B-splines978541 Node: Initializing the B-splines solver979850 Node: Constructing the knots vector981226 Node: Evaluation of B-spline basis functions982013 Node: Evaluation of B-spline basis function derivatives983670 Node: Obtaining Greville abscissae for B-spline basis functions985662 Node: Example programs for B-splines986700 Node: References and Further Reading990712 Node: Physical Constants991517 Node: Fundamental Constants992927 Node: Astronomy and Astrophysics994066 Node: Atomic and Nuclear Physics994731 Node: Measurement of Time996382 Node: Imperial Units996812 Node: Speed and Nautical Units997254 Node: Printers Units997758 Node: Volume Area and Length998081 Node: Mass and Weight998767 Node: Thermal Energy and Power999586 Node: Pressure1000009 Node: Viscosity1000622 Node: Light and Illumination1000898 Node: Radioactivity1001490 Node: Force and Energy1001825 Node: Prefixes1002229 Node: Physical Constant Examples1002972 Node: Physical Constant References and Further Reading1004761 Node: IEEE floating-point arithmetic1005491 Node: Representation of floating point numbers1006077 Node: Setting up your IEEE environment1010558 Node: IEEE References and Further Reading1017550 Node: Debugging Numerical Programs1018705 Node: Using gdb1019189 Node: Examining floating point registers1022532 Node: Handling floating point exceptions1023817 Node: GCC warning options for numerical programs1025229 Node: Debugging References1029191 Node: Contributors to GSL1029903 Node: Autoconf Macros1034138 Node: GSL CBLAS Library1038167 Node: Level 1 CBLAS Functions1038694 Node: Level 2 CBLAS Functions1043996 Node: Level 3 CBLAS Functions1060666 Node: GSL CBLAS Examples1070328 Node: Free Software Needs Free Documentation1071890 Node: GNU General Public License1076959 Node: GNU Free Documentation License1114571 Node: Function Index1139739 Node: Variable Index1384156 Node: Type Index1387459 Node: Concept Index1401858  End Tag Table