This is gsl-ref.info, produced by makeinfo version 4.8 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 Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 The GSL Team. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 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 freedom to copy and modify this GNU Manual, like GNU software."  Indirect: gsl-ref.info-1: 939 gsl-ref.info-2: 298842 gsl-ref.info-3: 596860 gsl-ref.info-4: 895867 gsl-ref.info-5: 1072596 gsl-ref.info-6: 1311271  Tag Table: (Indirect) Node: Top939 Node: Introduction3646 Node: Routines available in GSL4298 Node: GSL is Free Software6213 Node: Obtaining GSL8605 Node: No Warranty9746 Node: Reporting Bugs10252 Node: Further Information11136 Node: Conventions used in this manual12149 Node: Using the library12913 Node: An Example Program13499 Node: Compiling and Linking14203 Node: Linking programs with the library15273 Node: Linking with an alternative BLAS library16418 Node: Shared Libraries17348 Node: ANSI C Compliance18764 Node: Inline functions19857 Node: Long double20900 Node: Portability functions22298 Node: Alternative optimized functions23695 Node: Support for different numeric types25153 Node: Compatibility with C++28216 Node: Aliasing of arrays28788 Node: Thread-safety29539 Node: Deprecated Functions30626 Node: Code Reuse31251 Node: Error Handling31895 Node: Error Reporting32645 Node: Error Codes34537 Node: Error Handlers36393 Node: Using GSL error reporting in your own functions40028 Node: Error Reporting Examples41986 Node: Mathematical Functions43190 Node: Mathematical Constants43951 Node: Infinities and Not-a-number45081 Node: Elementary Functions46149 Node: Small integer powers48404 Node: Testing the Sign of Numbers49966 Node: Testing for Odd and Even Numbers50397 Node: Maximum and Minimum functions50953 Node: Approximate Comparison of Floating Point Numbers53035 Node: Complex Numbers54241 Ref: Complex Numbers-Footnote-155549 Node: Complex numbers55613 Node: Properties of complex numbers57416 Node: Complex arithmetic operators58403 Node: Elementary Complex Functions61149 Node: Complex Trigonometric Functions62969 Node: Inverse Complex Trigonometric Functions64190 Node: Complex Hyperbolic Functions66796 Node: Inverse Complex Hyperbolic Functions68092 Node: Complex Number References and Further Reading70080 Node: Polynomials71476 Node: Polynomial Evaluation72243 Node: Divided Difference Representation of Polynomials72703 Node: Quadratic Equations74372 Node: Cubic Equations76312 Node: General Polynomial Equations77683 Node: Roots of Polynomials Examples80028 Node: Roots of Polynomials References and Further Reading81417 Node: Special Functions82362 Node: Special Function Usage84356 Node: The gsl_sf_result struct85539 Node: Special Function Modes86803 Node: Airy Functions and Derivatives87739 Node: Airy Functions88435 Node: Derivatives of Airy Functions89795 Node: Zeros of Airy Functions91307 Node: Zeros of Derivatives of Airy Functions92026 Node: Bessel Functions92785 Node: Regular Cylindrical Bessel Functions93977 Node: Irregular Cylindrical Bessel Functions95301 Node: Regular Modified Cylindrical Bessel Functions96757 Node: Irregular Modified Cylindrical Bessel Functions99584 Node: Regular Spherical Bessel Functions102552 Node: Irregular Spherical Bessel Functions104764 Node: Regular Modified Spherical Bessel Functions106451 Node: Irregular Modified Spherical Bessel Functions108454 Node: Regular Bessel Function - Fractional Order110517 Node: Irregular Bessel Functions - Fractional Order111515 Node: Regular Modified Bessel Functions - Fractional Order112097 Node: Irregular Modified Bessel Functions - Fractional Order113022 Node: Zeros of Regular Bessel Functions114258 Node: Clausen Functions115361 Node: Coulomb Functions115965 Node: Normalized Hydrogenic Bound States116414 Node: Coulomb Wave Functions117527 Node: Coulomb Wave Function Normalization Constant120937 Node: Coupling Coefficients121690 Node: 3-j Symbols122413 Node: 6-j Symbols123017 Node: 9-j Symbols123641 Node: Dawson Function124347 Node: Debye Functions124914 Node: Dilogarithm126688 Node: Real Argument126983 Node: Complex Argument127552 Node: Elementary Operations128021 Node: Elliptic Integrals128845 Node: Definition of Legendre Forms129417 Node: Definition of Carlson Forms130237 Node: Legendre Form of Complete Elliptic Integrals130934 Node: Legendre Form of Incomplete Elliptic Integrals132502 Node: Carlson Forms134690 Node: Elliptic Functions (Jacobi)136237 Node: Error Functions136818 Node: Error Function137269 Node: Complementary Error Function137642 Node: Log Complementary Error Function138102 Node: Probability functions138554 Node: Exponential Functions139807 Node: Exponential Function140190 Node: Relative Exponential Functions141413 Node: Exponentiation With Error Estimate143084 Node: Exponential Integrals144293 Node: Exponential Integral144775 Node: Ei(x)145425 Node: Hyperbolic Integrals145873 Node: Ei_3(x)146529 Node: Trigonometric Integrals146921 Node: Arctangent Integral147520 Node: Fermi-Dirac Function147920 Node: Complete Fermi-Dirac Integrals148290 Node: Incomplete Fermi-Dirac Integrals150751 Node: Gamma and Beta Functions151394 Node: Gamma Functions151830 Node: Factorials154951 Node: Pochhammer Symbol157504 Node: Incomplete Gamma Functions159016 Node: Beta Functions160265 Node: Incomplete Beta Function161017 Node: Gegenbauer Functions161548 Node: Hypergeometric Functions163198 Node: Laguerre Functions167465 Node: Lambert W Functions169028 Node: Legendre Functions and Spherical Harmonics170066 Node: Legendre Polynomials170655 Node: Associated Legendre Polynomials and Spherical Harmonics172735 Node: Conical Functions175536 Node: Radial Functions for Hyperbolic Space177659 Node: Logarithm and Related Functions179836 Node: Mathieu Functions181460 Node: Mathieu Function Workspace182809 Node: Mathieu Function Characteristic Values183622 Node: Angular Mathieu Functions184690 Node: Radial Mathieu Functions185722 Node: Power Function186960 Node: Psi (Digamma) Function187887 Node: Digamma Function188407 Node: Trigamma Function189237 Node: Polygamma Function189814 Node: Synchrotron Functions190198 Node: Transport Functions190961 Node: Trigonometric Functions192108 Node: Circular Trigonometric Functions192731 Node: Trigonometric Functions for Complex Arguments193740 Node: Hyperbolic Trigonometric Functions194774 Node: Conversion Functions195377 Node: Restriction Functions196160 Node: Trigonometric Functions With Error Estimates197092 Node: Zeta Functions198002 Node: Riemann Zeta Function198460 Node: Riemann Zeta Function Minus One199139 Node: Hurwitz Zeta Function199910 Node: Eta Function200415 Node: Special Functions Examples200974 Node: Special Functions References and Further Reading202668 Node: Vectors and Matrices203775 Node: Data types204534 Node: Blocks205737 Node: Block allocation206653 Node: Reading and writing blocks208137 Node: Example programs for blocks210201 Node: Vectors210828 Node: Vector allocation212714 Node: Accessing vector elements214332 Node: Initializing vector elements216574 Node: Reading and writing vectors217268 Node: Vector views219358 Node: Copying vectors226800 Node: Exchanging elements227660 Node: Vector operations228218 Node: Finding maximum and minimum elements of vectors229732 Node: Vector properties231257 Node: Example programs for vectors231880 Node: Matrices234169 Node: Matrix allocation237027 Node: Accessing matrix elements238694 Node: Initializing matrix elements240402 Node: Reading and writing matrices241198 Node: Matrix views243301 Node: Creating row and column views250441 Node: Copying matrices253612 Node: Copying rows and columns254208 Node: Exchanging rows and columns255885 Node: Matrix operations257363 Node: Finding maximum and minimum elements of matrices259003 Node: Matrix properties260907 Node: Example programs for matrices261655 Node: Vector and Matrix References and Further Reading265726 Node: Permutations266212 Node: The Permutation struct267500 Node: Permutation allocation268004 Node: Accessing permutation elements269438 Node: Permutation properties270201 Node: Permutation functions270900 Node: Applying Permutations272157 Node: Reading and writing permutations274082 Node: Permutations in cyclic form276264 Node: Permutation Examples279689 Node: Permutation References and Further Reading282232 Node: Combinations282906 Node: The Combination struct283751 Node: Combination allocation284303 Node: Accessing combination elements286088 Node: Combination properties286667 Node: Combination functions287553 Node: Reading and writing combinations288493 Node: Combination Examples290693 Node: Combination References and Further Reading292001 Node: Sorting292405 Node: Sorting objects293355 Node: Sorting vectors296266 Node: Selecting the k smallest or largest elements298842 Node: Computing the rank302176 Node: Sorting Examples303377 Node: Sorting References and Further Reading305021 Node: BLAS Support305567 Node: GSL BLAS Interface308542 Node: Level 1 GSL BLAS Interface309029 Node: Level 2 GSL BLAS Interface315772 Node: Level 3 GSL BLAS Interface325066 Node: BLAS Examples336202 Node: BLAS References and Further Reading337623 Node: Linear Algebra338982 Node: LU Decomposition340164 Node: QR Decomposition345372 Node: QR Decomposition with Column Pivoting351174 Node: Singular Value Decomposition355584 Node: Cholesky Decomposition359174 Node: Tridiagonal Decomposition of Real Symmetric Matrices361181 Node: Tridiagonal Decomposition of Hermitian Matrices363029 Node: Hessenberg Decomposition of Real Matrices364984 Node: Bidiagonalization367329 Node: Householder Transformations369739 Node: Householder solver for linear systems371587 Node: Tridiagonal Systems372492 Node: Balancing375145 Node: Linear Algebra Examples375946 Node: Linear Algebra References and Further Reading377934 Node: Eigensystems379424 Node: Real Symmetric Matrices380603 Node: Complex Hermitian Matrices382835 Node: Real Nonsymmetric Matrices385343 Node: Sorting Eigenvalues and Eigenvectors390537 Node: Eigenvalue and Eigenvector Examples392404 Node: Eigenvalue and Eigenvector References397790 Node: Fast Fourier Transforms398498 Node: Mathematical Definitions399707 Node: Overview of complex data FFTs402112 Node: Radix-2 FFT routines for complex data405055 Node: Mixed-radix FFT routines for complex data409267 Node: Overview of real data FFTs418596 Node: Radix-2 FFT routines for real data421030 Node: Mixed-radix FFT routines for real data424037 Node: FFT References and Further Reading434617 Node: Numerical Integration437486 Node: Numerical Integration Introduction439031 Node: Integrands without weight functions440965 Node: Integrands with weight functions441794 Node: Integrands with singular weight functions442505 Node: QNG non-adaptive Gauss-Kronrod integration443423 Node: QAG adaptive integration444694 Node: QAGS adaptive integration with singularities447325 Node: QAGP adaptive integration with known singular points449125 Node: QAGI adaptive integration on infinite intervals450439 Node: QAWC adaptive integration for Cauchy principal values452732 Node: QAWS adaptive integration for singular functions453907 Node: QAWO adaptive integration for oscillatory functions457129 Node: QAWF adaptive integration for Fourier integrals460908 Node: Numerical integration error codes463533 Node: Numerical integration examples464301 Node: Numerical integration References and Further Reading466346 Node: Random Number Generation467075 Node: General comments on random numbers468577 Node: The Random Number Generator Interface470527 Node: Random number generator initialization471885 Node: Sampling from a random number generator473901 Node: Auxiliary random number generator functions477066 Node: Random number environment variables479386 Node: Copying random number generator state481912 Node: Reading and writing random number generator state482887 Node: Random number generator algorithms484299 Node: Unix random number generators494108 Node: Other random number generators497826 Node: Random Number Generator Performance506459 Node: Random Number Generator Examples507777 Node: Random Number References and Further Reading509333 Node: Random Number Acknowledgements510630 Node: Quasi-Random Sequences511116 Node: Quasi-random number generator initialization512223 Node: Sampling from a quasi-random number generator513238 Node: Auxiliary quasi-random number generator functions513855 Node: Saving and resorting quasi-random number generator state514801 Node: Quasi-random number generator algorithms515609 Node: Quasi-random number generator examples516384 Node: Quasi-random number references517369 Node: Random Number Distributions517895 Node: Random Number Distribution Introduction521094 Node: The Gaussian Distribution522889 Node: The Gaussian Tail Distribution525551 Node: The Bivariate Gaussian Distribution527220 Node: The Exponential Distribution528543 Node: The Laplace Distribution529682 Node: The Exponential Power Distribution530779 Node: The Cauchy Distribution532047 Node: The Rayleigh Distribution533296 Node: The Rayleigh Tail Distribution534468 Node: The Landau Distribution535340 Node: The Levy alpha-Stable Distributions536295 Node: The Levy skew alpha-Stable Distribution537353 Node: The Gamma Distribution538965 Node: The Flat (Uniform) Distribution540622 Node: The Lognormal Distribution541774 Node: The Chi-squared Distribution543122 Node: The F-distribution544526 Node: The t-distribution546162 Node: The Beta Distribution547556 Node: The Logistic Distribution548715 Node: The Pareto Distribution549844 Node: Spherical Vector Distributions551014 Node: The Weibull Distribution553848 Node: The Type-1 Gumbel Distribution555042 Node: The Type-2 Gumbel Distribution556279 Node: The Dirichlet Distribution557510 Node: General Discrete Distributions559178 Node: The Poisson Distribution563043 Node: The Bernoulli Distribution564051 Node: The Binomial Distribution564802 Node: The Multinomial Distribution566007 Node: The Negative Binomial Distribution567776 Node: The Pascal Distribution569141 Node: The Geometric Distribution570299 Node: The Hypergeometric Distribution571547 Node: The Logarithmic Distribution573200 Node: Shuffling and Sampling573991 Node: Random Number Distribution Examples576810 Node: Random Number Distribution References and Further Reading580006 Node: Statistics582072 Node: Mean and standard deviation and variance583351 Node: Absolute deviation586396 Node: Higher moments (skewness and kurtosis)587688 Node: Autocorrelation589821 Node: Covariance590635 Node: Weighted Samples591616 Node: Maximum and Minimum values596860 Node: Median and Percentiles599599 Node: Example statistical programs602014 Node: Statistics References and Further Reading604673 Node: Histograms605881 Node: The histogram struct607641 Node: Histogram allocation609444 Node: Copying Histograms612405 Node: Updating and accessing histogram elements613081 Node: Searching histogram ranges616348 Node: Histogram Statistics617347 Node: Histogram Operations619207 Node: Reading and writing histograms621279 Node: Resampling from histograms624321 Node: The histogram probability distribution struct625119 Node: Example programs for histograms628150 Node: Two dimensional histograms630213 Node: The 2D histogram struct630934 Node: 2D Histogram allocation632742 Node: Copying 2D Histograms634813 Node: Updating and accessing 2D histogram elements635518 Node: Searching 2D histogram ranges639173 Node: 2D Histogram Statistics640192 Node: 2D Histogram Operations643051 Node: Reading and writing 2D histograms645225 Node: Resampling from 2D histograms648852 Node: Example programs for 2D histograms651871 Node: N-tuples653699 Node: The ntuple struct654955 Node: Creating ntuples655433 Node: Opening an existing ntuple file656100 Node: Writing ntuples656728 Node: Reading ntuples657189 Node: Closing an ntuple file657520 Node: Histogramming ntuple values657860 Node: Example ntuple programs659866 Node: Ntuple References and Further Reading663195 Node: Monte Carlo Integration663516 Node: Monte Carlo Interface664761 Node: PLAIN Monte Carlo667384 Node: MISER669822 Node: VEGAS675660 Node: Monte Carlo Examples683995 Node: Monte Carlo Integration References and Further Reading689938 Node: Simulated Annealing690720 Node: Simulated Annealing algorithm691927 Node: Simulated Annealing functions693081 Node: Examples with Simulated Annealing697607 Node: Trivial example698161 Node: Traveling Salesman Problem700812 Node: Simulated Annealing References and Further Reading704125 Node: Ordinary Differential Equations704536 Node: Defining the ODE System705357 Node: Stepping Functions707549 Node: Adaptive Step-size Control711728 Node: Evolution717186 Node: ODE Example programs719864 Node: ODE References and Further Reading724934 Node: Interpolation725573 Node: Introduction to Interpolation726586 Node: Interpolation Functions727027 Node: Interpolation Types728203 Node: Index Look-up and Acceleration730813 Node: Evaluation of Interpolating Functions732443 Node: Higher-level Interface734827 Node: Interpolation Example programs736866 Node: Interpolation References and Further Reading740113 Node: Numerical Differentiation740686 Node: Numerical Differentiation functions741275 Node: Numerical Differentiation Examples744133 Node: Numerical Differentiation References745546 Node: Chebyshev Approximations746097 Node: Chebyshev Definitions747113 Node: Creation and Calculation of Chebyshev Series747903 Node: Chebyshev Series Evaluation748875 Node: Derivatives and Integrals750259 Node: Chebyshev Approximation examples751501 Node: Chebyshev Approximation References and Further Reading752997 Node: Series Acceleration753446 Node: Acceleration functions754211 Node: Acceleration functions without error estimation756536 Node: Example of accelerating a series759146 Node: Series Acceleration References761492 Node: Wavelet Transforms762380 Node: DWT Definitions762921 Node: DWT Initialization763873 Node: DWT Transform Functions766452 Node: DWT in one dimension766987 Node: DWT in two dimension769006 Node: DWT Examples773607 Node: DWT References775424 Node: Discrete Hankel Transforms777587 Node: Discrete Hankel Transform Definition778055 Node: Discrete Hankel Transform Functions780194 Node: Discrete Hankel Transform References781725 Node: One dimensional Root-Finding782129 Node: Root Finding Overview783389 Node: Root Finding Caveats785247 Node: Initializing the Solver787014 Node: Providing the function to solve789647 Node: Search Bounds and Guesses793413 Node: Root Finding Iteration794274 Node: Search Stopping Parameters796123 Node: Root Bracketing Algorithms798637 Node: Root Finding Algorithms using Derivatives801935 Node: Root Finding Examples805438 Node: Root Finding References and Further Reading812726 Node: One dimensional Minimization813365 Node: Minimization Overview814667 Node: Minimization Caveats816373 Node: Initializing the Minimizer817710 Node: Providing the function to minimize819951 Node: Minimization Iteration820429 Node: Minimization Stopping Parameters822565 Node: Minimization Algorithms824174 Node: Minimization Examples826613 Node: Minimization References and Further Reading829557 Node: Multidimensional Root-Finding830013 Node: Overview of Multidimensional Root Finding831502 Node: Initializing the Multidimensional Solver833689 Node: Providing the multidimensional system of equations to solve836887 Node: Iteration of the multidimensional solver841792 Node: Search Stopping Parameters for the multidimensional solver843977 Node: Algorithms using Derivatives845714 Node: Algorithms without Derivatives850460 Node: Example programs for Multidimensional Root finding853548 Node: References and Further Reading for Multidimensional Root Finding862161 Node: Multidimensional Minimization863400 Node: Multimin Overview864692 Node: Multimin Caveats866664 Node: Initializing the Multidimensional Minimizer867413 Node: Providing a function to minimize870574 Node: Multimin Iteration874239 Node: Multimin Stopping Criteria876230 Node: Multimin Algorithms877782 Node: Multimin Examples882741 Node: Multimin References and Further Reading889358 Node: Least-Squares Fitting890221 Node: Fitting Overview891220 Node: Linear regression893436 Node: Linear fitting without a constant term895867 Node: Multi-parameter fitting898058 Node: Fitting Examples902963 Node: Fitting References and Further Reading909788 Node: Nonlinear Least-Squares Fitting910610 Node: Overview of Nonlinear Least-Squares Fitting912060 Node: Initializing the Nonlinear Least-Squares Solver913527 Node: Providing the Function to be Minimized916380 Node: Iteration of the Minimization Algorithm919317 Node: Search Stopping Parameters for Minimization Algorithms921167 Node: Minimization Algorithms using Derivatives923262 Node: Minimization Algorithms without Derivatives926510 Node: Computing the covariance matrix of best fit parameters926902 Node: Example programs for Nonlinear Least-Squares Fitting928936 Node: References and Further Reading for Nonlinear Least-Squares Fitting936664 Node: Basis Splines937400 Node: Overview of B-splines937975 Node: Initializing the B-splines solver939260 Node: Constructing the knots vector940003 Node: Evaluation of B-spline basis functions940790 Node: Example programs for B-splines941523 Node: References and Further Reading945220 Node: Physical Constants945734 Node: Fundamental Constants947144 Node: Astronomy and Astrophysics948283 Node: Atomic and Nuclear Physics948948 Node: Measurement of Time950585 Node: Imperial Units951015 Node: Speed and Nautical Units951457 Node: Printers Units951961 Node: Volume Area and Length952284 Node: Mass and Weight952970 Node: Thermal Energy and Power953789 Node: Pressure954212 Node: Viscosity954825 Node: Light and Illumination955101 Node: Radioactivity955693 Node: Force and Energy956028 Node: Prefixes956432 Node: Physical Constant Examples957175 Node: Physical Constant References and Further Reading958964 Node: IEEE floating-point arithmetic959675 Node: Representation of floating point numbers960261 Node: Setting up your IEEE environment964742 Node: IEEE References and Further Reading971351 Node: Debugging Numerical Programs972506 Node: Using gdb972990 Node: Examining floating point registers976333 Node: Handling floating point exceptions977618 Node: GCC warning options for numerical programs979030 Node: Debugging References983223 Node: Contributors to GSL983935 Node: Autoconf Macros988118 Node: GSL CBLAS Library992103 Node: Level 1 CBLAS Functions992630 Node: Level 2 CBLAS Functions997932 Node: Level 3 CBLAS Functions1014602 Node: GSL CBLAS Examples1024264 Node: Free Software Needs Free Documentation1025826 Node: GNU General Public License1030895 Node: GNU Free Documentation License1050165 Node: Function Index1072596 Node: Variable Index1306453 Node: Type Index1307625 Node: Concept Index1311271  End Tag Table