This is gsl-ref.info, produced by makeinfo version 4.6 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 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: 929 gsl-ref.info-2: 297420 gsl-ref.info-3: 595737 gsl-ref.info-4: 895442 gsl-ref.info-5: 1158865  Tag Table: (Indirect) Node: Top929 Node: Introduction3495 Node: Routines available in GSL4143 Node: GSL is Free Software6050 Node: Obtaining GSL8434 Node: No Warranty9567 Node: Reporting Bugs10065 Node: Further Information10941 Node: Conventions used in this manual11946 Node: Using the library12702 Node: An Example Program13284 Node: Compiling and Linking13985 Node: Linking programs with the library15047 Node: Linking with an alternative BLAS library16180 Node: Shared Libraries17098 Node: ANSI C Compliance18506 Node: Inline functions19591 Node: Long double20626 Node: Portability functions22016 Node: Alternative optimized functions23410 Node: Support for different numeric types24860 Node: Compatibility with C++27915 Node: Aliasing of arrays28477 Node: Thread-safety29218 Node: Deprecated Functions30295 Node: Code Reuse30910 Node: Error Handling31544 Node: Error Reporting32290 Node: Error Codes34179 Node: Error Handlers35935 Node: Using GSL error reporting in your own functions39581 Node: Error Reporting Examples41529 Node: Mathematical Functions42747 Node: Mathematical Constants43504 Node: Infinities and Not-a-number44626 Node: Elementary Functions45680 Node: Small integer powers47919 Node: Testing the Sign of Numbers49461 Node: Testing for Odd and Even Numbers49883 Node: Maximum and Minimum functions50429 Node: Approximate Comparison of Floating Point Numbers52495 Node: Complex Numbers53694 Ref: Complex Numbers-Footnote-154998 Node: Complex numbers55062 Node: Properties of complex numbers56850 Node: Complex arithmetic operators57825 Node: Elementary Complex Functions60548 Node: Complex Trigonometric Functions62352 Node: Inverse Complex Trigonometric Functions63559 Node: Complex Hyperbolic Functions66147 Node: Inverse Complex Hyperbolic Functions67429 Node: Complex Number References and Further Reading69253 Node: Polynomials70641 Node: Polynomial Evaluation71404 Node: Divided Difference Representation of Polynomials71855 Node: Quadratic Equations73513 Node: Cubic Equations75443 Node: General Polynomial Equations76804 Node: Roots of Polynomials Examples79084 Node: Roots of Polynomials References and Further Reading80502 Node: Special Functions81439 Node: Special Function Usage83407 Node: The gsl_sf_result struct84582 Node: Special Function Modes85838 Node: Airy Functions and Derivatives86766 Node: Airy Functions87454 Node: Derivatives of Airy Functions88794 Node: Zeros of Airy Functions90286 Node: Zeros of Derivatives of Airy Functions90989 Node: Bessel Functions91732 Node: Regular Cylindrical Bessel Functions92916 Node: Irregular Cylindrical Bessel Functions94221 Node: Regular Modified Cylindrical Bessel Functions95658 Node: Irregular Modified Cylindrical Bessel Functions98459 Node: Regular Spherical Bessel Functions101401 Node: Irregular Spherical Bessel Functions103591 Node: Regular Modified Spherical Bessel Functions105257 Node: Irregular Modified Spherical Bessel Functions107239 Node: Regular Bessel Function - Fractional Order109281 Node: Irregular Bessel Functions - Fractional Order110264 Node: Regular Modified Bessel Functions - Fractional Order110830 Node: Irregular Modified Bessel Functions - Fractional Order111737 Node: Zeros of Regular Bessel Functions112953 Node: Clausen Functions114036 Node: Coulomb Functions114630 Node: Normalized Hydrogenic Bound States115071 Node: Coulomb Wave Functions116168 Node: Coulomb Wave Function Normalization Constant119565 Node: Coupling Coefficients120304 Node: 3-j Symbols121019 Node: 6-j Symbols121609 Node: 9-j Symbols122219 Node: Dawson Function122911 Node: Debye Functions123468 Node: Dilogarithm125221 Node: Real Argument125506 Node: Complex Argument126059 Node: Elementary Operations126513 Node: Elliptic Integrals127325 Node: Definition of Legendre Forms127788 Node: Definition of Carlson Forms128623 Node: Legendre Form of Complete Elliptic Integrals129321 Node: Legendre Form of Incomplete Elliptic Integrals130187 Node: Carlson Forms131905 Node: Elliptic Functions (Jacobi)133430 Node: Error Functions134000 Node: Error Function134441 Node: Complementary Error Function134798 Node: Log Complementary Error Function135242 Node: Probability functions135678 Node: Exponential Functions136911 Node: Exponential Function137284 Node: Relative Exponential Functions138477 Node: Exponentiation With Error Estimate140126 Node: Exponential Integrals141317 Node: Exponential Integral141789 Node: Ei(x)142419 Node: Hyperbolic Integrals142851 Node: Ei_3(x)143489 Node: Trigonometric Integrals143865 Node: Arctangent Integral144446 Node: Fermi-Dirac Function144830 Node: Complete Fermi-Dirac Integrals145190 Node: Incomplete Fermi-Dirac Integrals147621 Node: Gamma and Beta Functions148248 Node: Gamma Functions148674 Node: Factorials151738 Node: Pochhammer Symbol154263 Node: Incomplete Gamma Functions155733 Node: Beta Functions156962 Node: Incomplete Beta Function157631 Node: Gegenbauer Functions158137 Node: Hypergeometric Functions159768 Node: Laguerre Functions163969 Node: Lambert W Functions165514 Node: Legendre Functions and Spherical Harmonics166528 Node: Legendre Polynomials167107 Node: Associated Legendre Polynomials and Spherical Harmonics169107 Node: Conical Functions171887 Node: Radial Functions for Hyperbolic Space173984 Node: Logarithm and Related Functions176140 Node: Power Function177742 Node: Psi (Digamma) Function178671 Node: Digamma Function179181 Node: Trigamma Function179991 Node: Polygamma Function180550 Node: Synchrotron Functions180918 Node: Transport Functions181667 Node: Trigonometric Functions182756 Node: Circular Trigonometric Functions183369 Node: Trigonometric Functions for Complex Arguments184356 Node: Hyperbolic Trigonometric Functions185373 Node: Conversion Functions185958 Node: Restriction Functions186725 Node: Trigonometric Functions With Error Estimates187336 Node: Zeta Functions188230 Node: Riemann Zeta Function188678 Node: Riemann Zeta Function Minus One189339 Node: Hurwitz Zeta Function190092 Node: Eta Function190581 Node: Special Functions Examples191122 Node: Special Functions References and Further Reading192849 Node: Vectors and Matrices193945 Node: Data types194700 Node: Blocks195895 Node: Block allocation196803 Node: Reading and writing blocks198192 Node: Example programs for blocks200240 Node: Vectors200872 Node: Vector allocation202750 Node: Accessing vector elements204271 Node: Initializing vector elements206497 Node: Reading and writing vectors207176 Node: Vector views209240 Node: Copying vectors216658 Node: Exchanging elements217504 Node: Vector operations218048 Node: Finding maximum and minimum elements of vectors219544 Node: Vector properties221051 Node: Example programs for vectors221399 Node: Matrices223631 Node: Matrix allocation226481 Node: Accessing matrix elements228051 Node: Initializing matrix elements229743 Node: Reading and writing matrices230524 Node: Matrix views232601 Node: Creating row and column views239719 Node: Copying matrices242868 Node: Copying rows and columns243450 Node: Exchanging rows and columns245111 Node: Matrix operations246572 Node: Finding maximum and minimum elements of matrices248192 Node: Matrix properties250076 Node: Example programs for matrices250427 Node: Vector and Matrix References and Further Reading254476 Node: Permutations254954 Node: The Permutation struct256238 Node: Permutation allocation256733 Node: Accessing permutation elements258154 Node: Permutation properties258907 Node: Permutation functions259595 Node: Applying Permutations260840 Node: Reading and writing permutations262752 Node: Permutations in cyclic form264922 Node: Permutation Examples268334 Node: Permutation References and Further Reading270807 Node: Combinations271471 Node: The Combination struct272310 Node: Combination allocation272851 Node: Accessing combination elements274620 Node: Combination properties275188 Node: Combination functions276060 Node: Reading and writing combinations276988 Node: Combination Examples279174 Node: Combination References and Further Reading280490 Node: Sorting280883 Node: Sorting objects281827 Node: Sorting vectors284726 Node: Selecting the k smallest or largest elements287287 Node: Computing the rank290611 Node: Sorting Examples291817 Node: Sorting References and Further Reading293438 Node: BLAS Support293974 Node: GSL BLAS Interface296943 Node: Level 1 GSL BLAS Interface297420 Node: Level 2 GSL BLAS Interface304093 Node: Level 3 GSL BLAS Interface313348 Node: BLAS Examples324433 Node: BLAS References and Further Reading325887 Node: Linear Algebra327236 Node: LU Decomposition328484 Node: QR Decomposition333656 Node: QR Decomposition with Column Pivoting339415 Node: Singular Value Decomposition343807 Node: Cholesky Decomposition347159 Node: Tridiagonal Decomposition of Real Symmetric Matrices349067 Node: Tridiagonal Decomposition of Hermitian Matrices350902 Node: Bidiagonalization352820 Node: Householder Transformations355222 Node: Householder solver for linear systems357054 Node: Tridiagonal Systems357945 Node: Linear Algebra Examples360440 Node: Linear Algebra References and Further Reading362466 Node: Eigensystems363944 Node: Real Symmetric Matrices364995 Node: Complex Hermitian Matrices367211 Node: Sorting Eigenvalues and Eigenvectors369713 Node: Eigenvalue and Eigenvector Examples371032 Node: Eigenvalue and Eigenvector References373397 Node: Fast Fourier Transforms374095 Node: Mathematical Definitions375298 Node: Overview of complex data FFTs377693 Node: Radix-2 FFT routines for complex data380631 Node: Mixed-radix FFT routines for complex data384873 Node: Overview of real data FFTs394267 Node: Radix-2 FFT routines for real data396691 Node: Mixed-radix FFT routines for real data399517 Node: FFT References and Further Reading410166 Node: Numerical Integration413025 Node: Numerical Integration Introduction414564 Node: Integrands without weight functions416503 Node: Integrands with weight functions417318 Node: Integrands with singular weight functions418015 Node: QNG non-adaptive Gauss-Kronrod integration418919 Node: QAG adaptive integration420179 Node: QAGS adaptive integration with singularities422807 Node: QAGP adaptive integration with known singular points424596 Node: QAGI adaptive integration on infinite intervals425899 Node: QAWC adaptive integration for Cauchy principal values428169 Node: QAWS adaptive integration for singular functions429333 Node: QAWO adaptive integration for oscillatory functions432560 Node: QAWF adaptive integration for Fourier integrals436324 Node: Numerical integration error codes438936 Node: Numerical integration examples439692 Node: Numerical integration References and Further Reading441721 Node: Random Number Generation442437 Node: General comments on random numbers443933 Node: The Random Number Generator Interface445873 Node: Random number generator initialization447221 Node: Sampling from a random number generator449224 Node: Auxiliary random number generator functions452377 Node: Random number environment variables454711 Node: Copying random number generator state457228 Node: Reading and writing random number generator state458191 Node: Random number generator algorithms459591 Node: Unix random number generators469030 Node: Other random number generators472744 Node: Random Number Generator Performance482052 Node: Random Number Generator Examples483363 Node: Random Number References and Further Reading484945 Node: Random Number Acknowledgements486230 Node: Quasi-Random Sequences486704 Node: Quasi-random number generator initialization487805 Node: Sampling from a quasi-random number generator488807 Node: Auxiliary quasi-random number generator functions489413 Node: Saving and resorting quasi-random number generator state490346 Node: Quasi-random number generator algorithms491132 Node: Quasi-random number generator examples491895 Node: Quasi-random number references492885 Node: Random Number Distributions493401 Node: Random Number Distribution Introduction496594 Node: The Gaussian Distribution498379 Node: The Gaussian Tail Distribution501016 Node: The Bivariate Gaussian Distribution502661 Node: The Exponential Distribution503962 Node: The Laplace Distribution505085 Node: The Exponential Power Distribution506166 Node: The Cauchy Distribution507420 Node: The Rayleigh Distribution508653 Node: The Rayleigh Tail Distribution509809 Node: The Landau Distribution510667 Node: The Levy alpha-Stable Distributions511608 Node: The Levy skew alpha-Stable Distribution512653 Node: The Gamma Distribution514252 Node: The Flat (Uniform) Distribution515762 Node: The Lognormal Distribution516886 Node: The Chi-squared Distribution518216 Node: The F-distribution519602 Node: The t-distribution521210 Node: The Beta Distribution522586 Node: The Logistic Distribution523717 Node: The Pareto Distribution524828 Node: Spherical Vector Distributions525980 Node: The Weibull Distribution528798 Node: The Type-1 Gumbel Distribution529974 Node: The Type-2 Gumbel Distribution531193 Node: The Dirichlet Distribution532406 Node: General Discrete Distributions534059 Node: The Poisson Distribution537908 Node: The Bernoulli Distribution538890 Node: The Binomial Distribution539627 Node: The Multinomial Distribution540816 Node: The Negative Binomial Distribution542453 Node: The Pascal Distribution543802 Node: The Geometric Distribution544944 Node: The Hypergeometric Distribution546176 Node: The Logarithmic Distribution547813 Node: Shuffling and Sampling548590 Node: Random Number Distribution Examples551434 Node: Random Number Distribution References and Further Reading554669 Node: Statistics556723 Node: Mean and standard deviation and variance557996 Node: Absolute deviation561024 Node: Higher moments (skewness and kurtosis)562304 Node: Autocorrelation564423 Node: Covariance565225 Node: Weighted Samples566162 Node: Maximum and Minimum values571383 Node: Median and Percentiles574106 Node: Example statistical programs576509 Node: Statistics References and Further Reading579219 Node: Histograms580415 Node: The histogram struct582169 Node: Histogram allocation583961 Node: Copying Histograms586938 Node: Updating and accessing histogram elements587602 Node: Searching histogram ranges590851 Node: Histogram Statistics591839 Node: Histogram Operations593682 Node: Reading and writing histograms595737 Node: Resampling from histograms598755 Node: The histogram probability distribution struct599543 Node: Example programs for histograms602557 Node: Two dimensional histograms604633 Node: The 2D histogram struct605342 Node: 2D Histogram allocation607137 Node: Copying 2D Histograms609192 Node: Updating and accessing 2D histogram elements609883 Node: Searching 2D histogram ranges613514 Node: 2D Histogram Statistics614520 Node: 2D Histogram Operations617357 Node: Reading and writing 2D histograms619512 Node: Resampling from 2D histograms623153 Node: Example programs for 2D histograms626155 Node: N-tuples627953 Node: The ntuple struct629203 Node: Creating ntuples629671 Node: Opening an existing ntuple file630327 Node: Writing ntuples630944 Node: Reading ntuples631393 Node: Closing an ntuple file631713 Node: Histogramming ntuple values632042 Node: Example ntuple programs634037 Node: Ntuple References and Further Reading637478 Node: Monte Carlo Integration637789 Node: Monte Carlo Interface639028 Node: PLAIN Monte Carlo641665 Node: MISER644089 Node: VEGAS649908 Node: Monte Carlo Examples658220 Node: Monte Carlo Integration References and Further Reading664248 Node: Simulated Annealing665020 Node: Simulated Annealing algorithm666221 Node: Simulated Annealing functions667363 Node: Examples with Simulated Annealing671870 Node: Trivial example672414 Node: Traveling Salesman Problem675142 Node: Simulated Annealing References and Further Reading678478 Node: Ordinary Differential Equations678879 Node: Defining the ODE System679694 Node: Stepping Functions681875 Node: Adaptive Step-size Control685807 Node: Evolution691247 Node: ODE Example programs693794 Node: ODE References and Further Reading698978 Node: Interpolation699607 Node: Introduction to Interpolation700614 Node: Interpolation Functions701045 Node: Interpolation Types702208 Node: Index Look-up and Acceleration704800 Node: Evaluation of Interpolating Functions706416 Node: Higher-level Interface708782 Node: Interpolation Example programs710798 Node: Interpolation References and Further Reading714111 Node: Numerical Differentiation714673 Node: Numerical Differentiation functions715256 Node: Numerical Differentiation Examples718099 Node: Numerical Differentiation References719543 Node: Chebyshev Approximations720084 Node: Chebyshev Definitions721094 Node: Creation and Calculation of Chebyshev Series721874 Node: Chebyshev Series Evaluation722833 Node: Derivatives and Integrals724193 Node: Chebyshev Approximation examples725423 Node: Chebyshev Approximation References and Further Reading726956 Node: Series Acceleration727395 Node: Acceleration functions728154 Node: Acceleration functions without error estimation730383 Node: Example of accelerating a series732989 Node: Series Acceleration References735397 Node: Wavelet Transforms736275 Node: DWT Definitions736810 Node: DWT Initialization737752 Node: DWT Transform Functions740310 Node: DWT in one dimension740835 Node: DWT in two dimension742837 Node: DWT Examples747412 Node: DWT References749115 Node: Discrete Hankel Transforms751268 Node: Discrete Hankel Transform Definition751730 Node: Discrete Hankel Transform Functions753859 Node: Discrete Hankel Transform References755363 Node: One dimensional Root-Finding755757 Node: Root Finding Overview757011 Node: Root Finding Caveats758859 Node: Initializing the Solver760616 Node: Providing the function to solve763231 Node: Search Bounds and Guesses767030 Node: Root Finding Iteration767881 Node: Search Stopping Parameters769714 Node: Root Bracketing Algorithms772215 Node: Root Finding Algorithms using Derivatives775500 Node: Root Finding Examples778990 Node: Root Finding References and Further Reading786332 Node: One dimensional Minimization786959 Node: Minimization Overview788255 Node: Minimization Caveats789951 Node: Initializing the Minimizer791278 Node: Providing the function to minimize793504 Node: Minimization Iteration793972 Node: Minimization Stopping Parameters796091 Node: Minimization Algorithms797689 Node: Minimization Examples800116 Node: Minimization References and Further Reading803108 Node: Multidimensional Root-Finding803554 Node: Overview of Multidimensional Root Finding805037 Node: Initializing the Multidimensional Solver807214 Node: Providing the multidimensional system of equations to solve810384 Node: Iteration of the multidimensional solver815327 Node: Search Stopping Parameters for the multidimensional solver817494 Node: Algorithms using Derivatives819219 Node: Algorithms without Derivatives823951 Node: Example programs for Multidimensional Root finding826955 Node: References and Further Reading for Multidimensional Root Finding835743 Node: Multidimensional Minimization836972 Node: Multimin Overview838258 Node: Multimin Caveats840220 Node: Initializing the Multidimensional Minimizer840959 Node: Providing a function to minimize843864 Node: Multimin Iteration847562 Node: Multimin Stopping Criteria849534 Node: Multimin Algorithms851074 Node: Multimin Examples855387 Node: Multimin References and Further Reading862144 Node: Least-Squares Fitting862812 Node: Fitting Overview863805 Node: Linear regression865846 Node: Linear fitting without a constant term868264 Node: Multi-parameter fitting870442 Node: Fitting Examples875246 Node: Fitting References and Further Reading882063 Node: Nonlinear Least-Squares Fitting882876 Node: Overview of Nonlinear Least-Squares Fitting884324 Node: Initializing the Nonlinear Least-Squares Solver885781 Node: Providing the Function to be Minimized888609 Node: Iteration of the Minimization Algorithm891534 Node: Search Stopping Parameters for Minimization Algorithms893360 Node: Minimization Algorithms using Derivatives895442 Node: Minimization Algorithms without Derivatives898678 Node: Computing the covariance matrix of best fit parameters899060 Node: Example programs for Nonlinear Least-Squares Fitting901083 Node: References and Further Reading for Nonlinear Least-Squares Fitting908943 Node: Physical Constants909667 Node: Fundamental Constants911089 Node: Astronomy and Astrophysics912218 Node: Atomic and Nuclear Physics912873 Node: Measurement of Time914500 Node: Imperial Units914920 Node: Speed and Nautical Units915352 Node: Printers Units915846 Node: Volume Area and Length916159 Node: Mass and Weight916835 Node: Thermal Energy and Power917644 Node: Pressure918055 Node: Viscosity918656 Node: Light and Illumination918920 Node: Radioactivity919500 Node: Force and Energy919823 Node: Prefixes920215 Node: Physical Constant Examples920946 Node: Physical Constant References and Further Reading922755 Node: IEEE floating-point arithmetic923454 Node: Representation of floating point numbers924034 Node: Setting up your IEEE environment928547 Node: IEEE References and Further Reading935196 Node: Debugging Numerical Programs936340 Node: Using gdb936802 Node: Examining floating point registers940152 Node: Handling floating point exceptions941429 Node: GCC warning options for numerical programs942835 Node: Debugging References947020 Node: Contributors to GSL947724 Node: Autoconf Macros949930 Node: GSL CBLAS Library953908 Node: Level 1 CBLAS Functions954413 Node: Level 2 CBLAS Functions959661 Node: Level 3 CBLAS Functions976257 Node: GSL CBLAS Examples985880 Node: Free Software Needs Free Documentation987490 Node: GNU General Public License992559 Node: GNU Free Documentation License1011857 Node: Function Index1034293 Node: Variable Index1156126 Node: Type Index1156936 Node: Concept Index1158865  End Tag Table