This is gsl-ref.info, produced by makeinfo version 4.13 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, 2010, 2011 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: 948 gsl-ref.info-2: 300146 gsl-ref.info-3: 598317 gsl-ref.info-4: 898064 gsl-ref.info-5: 1168153 gsl-ref.info-6: 1435602  Tag Table: (Indirect) Node: Top948 Node: Introduction3679 Node: Routines available in GSL4331 Node: GSL is Free Software6246 Node: Obtaining GSL8638 Node: No Warranty9665 Node: Reporting Bugs10171 Ref: Reporting Bugs-Footnote-111127 Node: Further Information11177 Node: Conventions used in this manual12190 Node: Using the library12954 Node: An Example Program13540 Ref: An Example Program-Footnote-114283 Node: Compiling and Linking14390 Node: Linking programs with the library15460 Ref: Linking programs with the library-Footnote-116862 Ref: Linking programs with the library-Footnote-216899 Node: Linking with an alternative BLAS library16952 Node: Shared Libraries18034 Ref: Shared Libraries-Footnote-119509 Node: ANSI C Compliance19557 Node: Inline functions20650 Node: Long double22188 Node: Portability functions23776 Node: Alternative optimized functions25173 Node: Support for different numeric types26631 Node: Compatibility with C++29694 Node: Aliasing of arrays30266 Node: Thread-safety31017 Node: Deprecated Functions32104 Node: Code Reuse32729 Node: Error Handling33373 Node: Error Reporting34123 Node: Error Codes36015 Node: Error Handlers37871 Node: Using GSL error reporting in your own functions41508 Node: Error Reporting Examples43466 Node: Mathematical Functions44678 Node: Mathematical Constants45439 Node: Infinities and Not-a-number46569 Ref: Infinities and Not-a-number-Footnote-147676 Node: Elementary Functions48086 Node: Small integer powers50533 Node: Testing the Sign of Numbers52180 Node: Testing for Odd and Even Numbers52611 Node: Maximum and Minimum functions53167 Node: Approximate Comparison of Floating Point Numbers55442 Node: Complex Numbers56801 Ref: Complex Numbers-Footnote-158127 Node: Representation of complex numbers58191 Node: Properties of complex numbers60128 Node: Complex arithmetic operators61133 Node: Elementary Complex Functions63879 Node: Complex Trigonometric Functions65699 Node: Inverse Complex Trigonometric Functions66920 Node: Complex Hyperbolic Functions69526 Node: Inverse Complex Hyperbolic Functions70822 Node: Complex Number References and Further Reading72810 Node: Polynomials74206 Node: Polynomial Evaluation74973 Node: Divided Difference Representation of Polynomials76383 Node: Quadratic Equations78132 Node: Cubic Equations80072 Node: General Polynomial Equations81654 Node: Roots of Polynomials Examples83999 Node: Roots of Polynomials References and Further Reading85388 Node: Special Functions86333 Node: Special Function Usage88327 Node: The gsl_sf_result struct89510 Node: Special Function Modes90774 Node: Airy Functions and Derivatives91710 Node: Airy Functions92411 Node: Derivatives of Airy Functions93771 Node: Zeros of Airy Functions95283 Node: Zeros of Derivatives of Airy Functions96002 Node: Bessel Functions96761 Node: Regular Cylindrical Bessel Functions97953 Node: Irregular Cylindrical Bessel Functions99277 Node: Regular Modified Cylindrical Bessel Functions100735 Node: Irregular Modified Cylindrical Bessel Functions103562 Node: Regular Spherical Bessel Functions106530 Node: Irregular Spherical Bessel Functions108744 Node: Regular Modified Spherical Bessel Functions110431 Node: Irregular Modified Spherical Bessel Functions112434 Node: Regular Bessel Function - Fractional Order114497 Node: Irregular Bessel Functions - Fractional Order115495 Node: Regular Modified Bessel Functions - Fractional Order116077 Node: Irregular Modified Bessel Functions - Fractional Order117002 Node: Zeros of Regular Bessel Functions118238 Node: Clausen Functions119341 Node: Coulomb Functions119945 Node: Normalized Hydrogenic Bound States120394 Node: Coulomb Wave Functions121507 Node: Coulomb Wave Function Normalization Constant124917 Node: Coupling Coefficients125670 Node: 3-j Symbols126393 Node: 6-j Symbols126997 Node: 9-j Symbols127621 Node: Dawson Function128327 Node: Debye Functions128894 Node: Dilogarithm130668 Node: Real Argument130963 Node: Complex Argument131656 Node: Elementary Operations132125 Node: Elliptic Integrals132949 Node: Definition of Legendre Forms133521 Node: Definition of Carlson Forms134341 Node: Legendre Form of Complete Elliptic Integrals135038 Node: Legendre Form of Incomplete Elliptic Integrals136606 Node: Carlson Forms138794 Node: Elliptic Functions (Jacobi)140341 Node: Error Functions140922 Node: Error Function141373 Node: Complementary Error Function141746 Node: Log Complementary Error Function142206 Node: Probability functions142658 Node: Exponential Functions143911 Node: Exponential Function144294 Node: Relative Exponential Functions145517 Node: Exponentiation With Error Estimate147189 Node: Exponential Integrals148398 Node: Exponential Integral148880 Node: Ei(x)149807 Node: Hyperbolic Integrals150255 Node: Ei_3(x)150911 Node: Trigonometric Integrals151303 Node: Arctangent Integral151902 Node: Fermi-Dirac Function152302 Node: Complete Fermi-Dirac Integrals152672 Node: Incomplete Fermi-Dirac Integrals155240 Node: Gamma and Beta Functions155883 Node: Gamma Functions156456 Node: Factorials159565 Node: Pochhammer Symbol162118 Node: Incomplete Gamma Functions163624 Node: Beta Functions164873 Node: Incomplete Beta Function165625 Node: Gegenbauer Functions166336 Node: Hypergeometric Functions167986 Node: Laguerre Functions172249 Node: Lambert W Functions173812 Node: Legendre Functions and Spherical Harmonics174850 Node: Legendre Polynomials175439 Node: Associated Legendre Polynomials and Spherical Harmonics177494 Node: Conical Functions180327 Node: Radial Functions for Hyperbolic Space182450 Node: Logarithm and Related Functions184627 Node: Mathieu Functions186251 Node: Mathieu Function Workspace187600 Node: Mathieu Function Characteristic Values188414 Node: Angular Mathieu Functions189486 Node: Radial Mathieu Functions190522 Node: Power Function191764 Node: Psi (Digamma) Function192691 Node: Digamma Function193211 Node: Trigamma Function194041 Node: Polygamma Function194618 Node: Synchrotron Functions195002 Node: Transport Functions195765 Node: Trigonometric Functions196912 Node: Circular Trigonometric Functions197535 Node: Trigonometric Functions for Complex Arguments198544 Node: Hyperbolic Trigonometric Functions199580 Node: Conversion Functions200183 Node: Restriction Functions200966 Node: Trigonometric Functions With Error Estimates201898 Node: Zeta Functions202808 Node: Riemann Zeta Function203266 Node: Riemann Zeta Function Minus One203945 Node: Hurwitz Zeta Function204716 Node: Eta Function205221 Node: Special Functions Examples205780 Node: Special Functions References and Further Reading207474 Node: Vectors and Matrices208744 Node: Data types209503 Node: Blocks210706 Node: Block allocation211622 Node: Reading and writing blocks213106 Node: Example programs for blocks215170 Node: Vectors215797 Node: Vector allocation217683 Node: Accessing vector elements219301 Ref: Accessing vector elements-Footnote-1222443 Node: Initializing vector elements222682 Node: Reading and writing vectors223376 Node: Vector views225466 Node: Copying vectors232908 Node: Exchanging elements233768 Node: Vector operations234326 Node: Finding maximum and minimum elements of vectors236092 Node: Vector properties237678 Node: Example programs for vectors238652 Node: Matrices240941 Node: Matrix allocation243799 Node: Accessing matrix elements245466 Node: Initializing matrix elements247415 Node: Reading and writing matrices248211 Node: Matrix views250314 Node: Creating row and column views257454 Node: Copying matrices261924 Node: Copying rows and columns262520 Node: Exchanging rows and columns264197 Node: Matrix operations265675 Node: Finding maximum and minimum elements of matrices267642 Node: Matrix properties269546 Node: Example programs for matrices270648 Node: Vector and Matrix References and Further Reading274719 Node: Permutations275205 Node: The Permutation struct276493 Node: Permutation allocation276997 Node: Accessing permutation elements278431 Node: Permutation properties279274 Node: Permutation functions279979 Node: Applying Permutations281236 Node: Reading and writing permutations283162 Ref: Reading and writing permutations-Footnote-1285377 Node: Permutations in cyclic form285489 Node: Permutation Examples288914 Node: Permutation References and Further Reading291457 Node: Combinations292131 Node: The Combination struct292978 Node: Combination allocation293530 Node: Accessing combination elements295315 Node: Combination properties295974 Node: Combination functions296860 Node: Reading and writing combinations297800 Ref: Reading and writing combinations-Footnote-1300034 Node: Combination Examples300146 Node: Combination References and Further Reading301454 Node: Multisets301858 Node: The Multiset struct302784 Node: Multiset allocation303306 Node: Accessing multiset elements304997 Node: Multiset properties305626 Node: Multiset functions306439 Node: Reading and writing multisets307349 Ref: Reading and writing multisets-Footnote-1309500 Node: Multiset Examples309612 Node: Sorting311720 Node: Sorting objects312667 Node: Sorting vectors315578 Node: Selecting the k smallest or largest elements318154 Node: Computing the rank321488 Node: Sorting Examples322689 Node: Sorting References and Further Reading324333 Node: BLAS Support324879 Ref: BLAS Support-Footnote-1328254 Node: GSL BLAS Interface328422 Node: Level 1 GSL BLAS Interface328909 Node: Level 2 GSL BLAS Interface335648 Node: Level 3 GSL BLAS Interface344944 Node: BLAS Examples356084 Node: BLAS References and Further Reading357505 Node: Linear Algebra358852 Node: LU Decomposition359977 Node: QR Decomposition365222 Node: QR Decomposition with Column Pivoting371531 Node: Singular Value Decomposition375986 Node: Cholesky Decomposition379982 Node: Tridiagonal Decomposition of Real Symmetric Matrices383222 Node: Tridiagonal Decomposition of Hermitian Matrices385070 Node: Hessenberg Decomposition of Real Matrices387022 Node: Hessenberg-Triangular Decomposition of Real Matrices389432 Node: Bidiagonalization390552 Node: Householder Transformations392977 Node: Householder solver for linear systems395329 Node: Tridiagonal Systems396234 Node: Balancing399106 Node: Linear Algebra Examples399907 Node: Linear Algebra References and Further Reading401895 Node: Eigensystems403542 Node: Real Symmetric Matrices404756 Node: Complex Hermitian Matrices407270 Node: Real Nonsymmetric Matrices409897 Node: Real Generalized Symmetric-Definite Eigensystems415610 Node: Complex Generalized Hermitian-Definite Eigensystems418482 Node: Real Generalized Nonsymmetric Eigensystems421189 Node: Sorting Eigenvalues and Eigenvectors427773 Node: Eigenvalue and Eigenvector Examples431050 Node: Eigenvalue and Eigenvector References436440 Node: Fast Fourier Transforms437465 Node: Mathematical Definitions438674 Node: Overview of complex data FFTs441079 Node: Radix-2 FFT routines for complex data444022 Node: Mixed-radix FFT routines for complex data448234 Node: Overview of real data FFTs457563 Node: Radix-2 FFT routines for real data459997 Node: Mixed-radix FFT routines for real data464426 Node: FFT References and Further Reading475100 Node: Numerical Integration477969 Node: Numerical Integration Introduction479735 Node: Integrands without weight functions482004 Node: Integrands with weight functions482833 Node: Integrands with singular weight functions483544 Node: QNG non-adaptive Gauss-Kronrod integration484462 Node: QAG adaptive integration485743 Node: QAGS adaptive integration with singularities488374 Node: QAGP adaptive integration with known singular points490174 Node: QAGI adaptive integration on infinite intervals491488 Node: QAWC adaptive integration for Cauchy principal values493781 Node: QAWS adaptive integration for singular functions494956 Node: QAWO adaptive integration for oscillatory functions498178 Node: QAWF adaptive integration for Fourier integrals501957 Node: CQUAD doubly-adaptive integration504582 Node: Fixed order Gauss-Legendre integration507819 Node: Numerical integration error codes509797 Node: Numerical integration examples510556 Node: Numerical integration References and Further Reading512602 Node: Random Number Generation513593 Node: General comments on random numbers515095 Node: The Random Number Generator Interface517045 Node: Random number generator initialization518403 Node: Sampling from a random number generator520776 Node: Auxiliary random number generator functions524017 Node: Random number environment variables526337 Node: Copying random number generator state528863 Node: Reading and writing random number generator state529838 Node: Random number generator algorithms531232 Node: Unix random number generators541167 Node: Other random number generators544885 Node: Random Number Generator Performance553519 Node: Random Number Generator Examples554640 Node: Random Number References and Further Reading556196 Node: Random Number Acknowledgements557493 Node: Quasi-Random Sequences557979 Node: Quasi-random number generator initialization559086 Node: Sampling from a quasi-random number generator560101 Node: Auxiliary quasi-random number generator functions560798 Node: Saving and resorting quasi-random number generator state561744 Node: Quasi-random number generator algorithms562552 Node: Quasi-random number generator examples563688 Node: Quasi-random number references564673 Node: Random Number Distributions565199 Node: Random Number Distribution Introduction568398 Node: The Gaussian Distribution570193 Node: The Gaussian Tail Distribution572854 Node: The Bivariate Gaussian Distribution574523 Node: The Exponential Distribution575846 Node: The Laplace Distribution576985 Node: The Exponential Power Distribution578082 Node: The Cauchy Distribution579350 Node: The Rayleigh Distribution580599 Node: The Rayleigh Tail Distribution581771 Node: The Landau Distribution582643 Node: The Levy alpha-Stable Distributions583598 Node: The Levy skew alpha-Stable Distribution584656 Node: The Gamma Distribution586268 Node: The Flat (Uniform) Distribution587925 Node: The Lognormal Distribution589077 Node: The Chi-squared Distribution590425 Node: The F-distribution591829 Node: The t-distribution593465 Node: The Beta Distribution594859 Node: The Logistic Distribution596018 Node: The Pareto Distribution597147 Node: Spherical Vector Distributions598317 Node: The Weibull Distribution601151 Node: The Type-1 Gumbel Distribution602345 Node: The Type-2 Gumbel Distribution603582 Node: The Dirichlet Distribution604813 Node: General Discrete Distributions606480 Node: The Poisson Distribution610347 Node: The Bernoulli Distribution611355 Node: The Binomial Distribution612106 Node: The Multinomial Distribution613311 Node: The Negative Binomial Distribution615080 Node: The Pascal Distribution616445 Node: The Geometric Distribution617603 Node: The Hypergeometric Distribution618851 Node: The Logarithmic Distribution620504 Node: Shuffling and Sampling621295 Node: Random Number Distribution Examples624114 Node: Random Number Distribution References and Further Reading627310 Node: Statistics629453 Node: Mean and standard deviation and variance630853 Node: Absolute deviation634361 Node: Higher moments (skewness and kurtosis)635653 Node: Autocorrelation637786 Node: Covariance638600 Node: Correlation639576 Node: Weighted Samples640285 Node: Maximum and Minimum values646135 Node: Median and Percentiles648874 Node: Example statistical programs651289 Node: Statistics References and Further Reading653949 Node: Histograms655157 Node: The histogram struct656917 Node: Histogram allocation658720 Node: Copying Histograms661681 Node: Updating and accessing histogram elements662357 Node: Searching histogram ranges665624 Node: Histogram Statistics666623 Node: Histogram Operations668483 Node: Reading and writing histograms670555 Node: Resampling from histograms673597 Node: The histogram probability distribution struct674395 Node: Example programs for histograms677426 Node: Two dimensional histograms679489 Node: The 2D histogram struct680210 Node: 2D Histogram allocation682018 Node: Copying 2D Histograms684089 Node: Updating and accessing 2D histogram elements684794 Node: Searching 2D histogram ranges688450 Node: 2D Histogram Statistics689469 Node: 2D Histogram Operations692328 Node: Reading and writing 2D histograms694502 Node: Resampling from 2D histograms698131 Node: Example programs for 2D histograms701150 Node: N-tuples702978 Node: The ntuple struct704234 Node: Creating ntuples704712 Node: Opening an existing ntuple file705379 Node: Writing ntuples706007 Node: Reading ntuples706468 Node: Closing an ntuple file706799 Node: Histogramming ntuple values707139 Node: Example ntuple programs709147 Node: Ntuple References and Further Reading712476 Node: Monte Carlo Integration712797 Node: Monte Carlo Interface714042 Node: PLAIN Monte Carlo716665 Node: MISER719119 Ref: MISER-Footnote-1725795 Node: VEGAS725920 Node: Monte Carlo Examples735471 Node: Monte Carlo Integration References and Further Reading741456 Node: Simulated Annealing742238 Node: Simulated Annealing algorithm743445 Node: Simulated Annealing functions744599 Node: Examples with Simulated Annealing749134 Node: Trivial example749688 Node: Traveling Salesman Problem752339 Node: Simulated Annealing References and Further Reading755652 Node: Ordinary Differential Equations756063 Node: Defining the ODE System757254 Node: Stepping Functions760080 Node: Adaptive Step-size Control767022 Node: Evolution773094 Node: Driver777211 Node: ODE Example programs781150 Node: ODE References and Further Reading786218 Node: Interpolation787978 Node: Introduction to Interpolation789149 Node: Interpolation Functions789590 Node: Interpolation Types790797 Node: Index Look-up and Acceleration793520 Node: Evaluation of Interpolating Functions795572 Node: Higher-level Interface798073 Node: Interpolation Example programs800112 Node: Interpolation References and Further Reading803328 Node: Numerical Differentiation803901 Node: Numerical Differentiation functions804490 Node: Numerical Differentiation Examples807348 Node: Numerical Differentiation References808761 Node: Chebyshev Approximations809312 Node: Chebyshev Definitions810373 Node: Creation and Calculation of Chebyshev Series811163 Node: Auxiliary Functions for Chebyshev Series812148 Node: Chebyshev Series Evaluation812888 Node: Derivatives and Integrals814268 Node: Chebyshev Approximation Examples815510 Node: Chebyshev Approximation References and Further Reading817006 Node: Series Acceleration817455 Node: Acceleration functions818220 Node: Acceleration functions without error estimation820545 Node: Example of accelerating a series823155 Node: Series Acceleration References825501 Node: Wavelet Transforms826389 Node: DWT Definitions826930 Node: DWT Initialization827882 Node: DWT Transform Functions830531 Node: DWT in one dimension831066 Node: DWT in two dimension833085 Node: DWT Examples837663 Node: DWT References839480 Node: Discrete Hankel Transforms841643 Node: Discrete Hankel Transform Definition842111 Node: Discrete Hankel Transform Functions844310 Node: Discrete Hankel Transform References845974 Node: One dimensional Root-Finding846378 Node: Root Finding Overview847638 Node: Root Finding Caveats849496 Node: Initializing the Solver851263 Node: Providing the function to solve853896 Node: Search Bounds and Guesses857444 Node: Root Finding Iteration858307 Node: Search Stopping Parameters860156 Node: Root Bracketing Algorithms862670 Node: Root Finding Algorithms using Derivatives865968 Ref: Root Finding Algorithms using Derivatives-Footnote-1869510 Node: Root Finding Examples869665 Node: Root Finding References and Further Reading876953 Node: One dimensional Minimization877592 Node: Minimization Overview878894 Node: Minimization Caveats880600 Node: Initializing the Minimizer881937 Node: Providing the function to minimize884178 Node: Minimization Iteration884656 Node: Minimization Stopping Parameters886792 Node: Minimization Algorithms888401 Node: Minimization Examples891005 Node: Minimization References and Further Reading893932 Node: Multidimensional Root-Finding894388 Node: Overview of Multidimensional Root Finding895877 Node: Initializing the Multidimensional Solver898064 Node: Providing the multidimensional system of equations to solve901295 Node: Iteration of the multidimensional solver906200 Node: Search Stopping Parameters for the multidimensional solver908479 Node: Algorithms using Derivatives910216 Node: Algorithms without Derivatives915045 Node: Example programs for Multidimensional Root finding918133 Node: References and Further Reading for Multidimensional Root Finding926746 Node: Multidimensional Minimization927985 Node: Multimin Overview929338 Node: Multimin Caveats931415 Node: Initializing the Multidimensional Minimizer932164 Node: Providing a function to minimize935325 Node: Multimin Iteration939377 Node: Multimin Stopping Criteria941576 Node: Multimin Algorithms with Derivatives943145 Node: Multimin Algorithms without Derivatives946673 Node: Multimin Examples949673 Node: Multimin References and Further Reading956239 Node: Least-Squares Fitting957104 Node: Fitting Overview958103 Node: Linear regression960437 Node: Linear fitting without a constant term963039 Node: Multi-parameter fitting965232 Node: Fitting Examples971046 Node: Fitting References and Further Reading977871 Node: Nonlinear Least-Squares Fitting978693 Node: Overview of Nonlinear Least-Squares Fitting980143 Node: Initializing the Nonlinear Least-Squares Solver981610 Node: Providing the Function to be Minimized984475 Node: Iteration of the Minimization Algorithm987412 Node: Search Stopping Parameters for Minimization Algorithms989262 Node: Minimization Algorithms using Derivatives991357 Ref: Minimization Algorithms using Derivatives-Footnote-1994838 Node: Minimization Algorithms without Derivatives994927 Node: Computing the covariance matrix of best fit parameters995319 Node: Example programs for Nonlinear Least-Squares Fitting997353 Node: References and Further Reading for Nonlinear Least-Squares Fitting1005081 Node: Basis Splines1005817 Node: Overview of B-splines1006717 Node: Initializing the B-splines solver1008026 Node: Constructing the knots vector1009402 Node: Evaluation of B-spline basis functions1010189 Node: Evaluation of B-spline basis function derivatives1011846 Node: Obtaining Greville abscissae for B-spline basis functions1013838 Node: Example programs for B-splines1014880 Node: References and Further Reading1018892 Node: Physical Constants1019697 Node: Fundamental Constants1021107 Node: Astronomy and Astrophysics1022246 Node: Atomic and Nuclear Physics1022911 Node: Measurement of Time1024562 Node: Imperial Units1024992 Node: Speed and Nautical Units1025434 Node: Printers Units1025938 Node: Volume Area and Length1026261 Node: Mass and Weight1026947 Node: Thermal Energy and Power1027766 Node: Pressure1028189 Node: Viscosity1028802 Node: Light and Illumination1029078 Node: Radioactivity1029670 Node: Force and Energy1030005 Node: Prefixes1030409 Node: Physical Constant Examples1031152 Node: Physical Constant References and Further Reading1032941 Node: IEEE floating-point arithmetic1033671 Node: Representation of floating point numbers1034257 Node: Setting up your IEEE environment1038738 Node: IEEE References and Further Reading1045730 Node: Debugging Numerical Programs1046885 Node: Using gdb1047369 Node: Examining floating point registers1050712 Node: Handling floating point exceptions1051997 Node: GCC warning options for numerical programs1053409 Node: Debugging References1057371 Node: Contributors to GSL1058083 Node: Autoconf Macros1062552 Node: GSL CBLAS Library1066581 Node: Level 1 CBLAS Functions1067108 Node: Level 2 CBLAS Functions1072410 Node: Level 3 CBLAS Functions1089080 Node: GSL CBLAS Examples1098742 Node: Free Software Needs Free Documentation1100304 Node: GNU General Public License1105373 Node: GNU Free Documentation License1142985 Node: Function Index1168153 Node: Variable Index1417524 Node: Type Index1420827 Node: Concept Index1435602  End Tag Table