# Concept Index

## \$

• `\$`, shell prompt
• ## 2

• 2D histograms
• 2D random direction vector
• ## 3

• 3-j symbols
• 3D random direction vector

• 6-j symbols

• 9-j symbols
• ## a

• acceleration of series
• acosh
• Adaptive step-size control, differential equations
• Ai(x)
• Airy functions
• Akima splines
• aliasing of arrays
• alternative optimized functions
• AMAX, Level-1 BLAS
• Angular Mathieu Functions
• angular reduction
• ANSI C, use of
• Apell symbol, see Pochhammer symbol
• approximate comparison of floating point numbers
• arctangent integral
• argument of complex number
• arithmetic exceptions
• asinh
• astronomical constants
• ASUM, Level-1 BLAS
• atanh
• atomic physics, constants
• autoconf, using with GSL
• AXPY, Level-1 BLAS
• ## b

• B-spline wavelets
• Bader and Deuflhard, Bulirsch-Stoer method.
• balancing matrices
• Basic Linear Algebra Subroutines (BLAS), Basic Linear Algebra Subroutines (BLAS)
• basis splines, B-splines
• basis splines, derivatives
• basis splines, evaluation
• basis splines, examples
• basis splines, Greville abscissae
• basis splines, initializing
• basis splines, Marsden-Schoenberg points
• basis splines, overview
• BDF method
• Bernoulli trial, random variates
• Bessel functions
• Bessel Functions, Fractional Order
• best-fit parameters, covariance
• Beta distribution
• Beta function
• Beta function, incomplete normalized
• BFGS algorithm, minimization
• Bi(x)
• bias, IEEE format
• bidiagonalization of real matrices
• binning data
• Binomial random variates
• biorthogonal wavelets
• bisection algorithm for finding roots
• Bivariate Gaussian distribution
• BLAS
• BLAS, Low-level C interface
• blocks
• bounds checking, extension to GCC
• breakpoints
• Brent's method for finding minima
• Brent's method for finding roots
• Broyden algorithm for multidimensional roots
• BSD random number generator
• bug-gsl mailing list
• bugs, how to report
• Bulirsch-Stoer method
• ## c

• C extensions, compatible use of
• C++, compatibility
• C99, inline keyword
• Carlson forms of Elliptic integrals
• Cash-Karp, Runge-Kutta method
• Cauchy distribution
• Cauchy principal value, by numerical quadrature
• CBLAS
• CBLAS, Low-level interface
• CDFs, cumulative distribution functions
• ce(q,x), Mathieu function
• Chebyshev series
• checking combination for validity
• checking multiset for validity
• checking permutation for validity
• Chi(x)
• Chi-squared distribution
• Cholesky decomposition
• Ci(x)
• Clausen functions
• CMRG, combined multiple recursive random number generator
• code reuse in applications
• combinations
• combinatorial factor C(m,n)
• combinatorial optimization
• comparison functions, definition
• compatibility
• compiling programs, include paths
• compiling programs, library paths
• complementary incomplete Gamma function
• complete Fermi-Dirac integrals
• complex arithmetic
• complex cosine function, special functions
• Complex Gamma function
• complex hermitian matrix, eigensystem
• complex log sine function, special functions
• complex numbers
• complex sinc function, special functions
• complex sine function, special functions
• confluent hypergeometric function
• confluent hypergeometric functions
• conical functions
• conjugate of complex number
• constant matrix
• constants, fundamental
• constants, mathematical--defined as macros
• constants, physical
• constants, prefixes
• contacting the GSL developers
• conventions, used in manual
• convergence, accelerating a series
• conversion of units
• cooling schedule
• COPY, Level-1 BLAS
• correlation, of two datasets
• cosine function, special functions
• cosine of complex number
• cost function
• Coulomb wave functions
• coupling coefficients
• covariance matrix, from linear regression
• covariance matrix, linear fits
• covariance matrix, nonlinear fits
• covariance, of two datasets
• CRAY random number generator, RANF
• cubic equation, solving
• cubic splines
• cumulative distribution functions (CDFs)
• Cylindrical Bessel Functions
• ## d

• Daubechies wavelets
• Dawson function
• DAXPY, Level-1 BLAS
• debugging numerical programs
• Debye functions
• denormalized form, IEEE format
• deprecated functions
• derivatives, calculating numerically
• determinant of a matrix, by LU decomposition
• Deuflhard and Bader, Bulirsch-Stoer method.
• DFTs, see FFT
• diagonal, of a matrix
• differential equations, initial value problems
• differentiation of functions, numeric
• digamma function
• dilogarithm
• direction vector, random 2D
• direction vector, random 3D
• direction vector, random N-dimensional
• Dirichlet distribution
• discontinuities, in ODE systems
• Discrete Fourier Transforms, see FFT
• discrete Hankel transforms
• Discrete Newton algorithm for multidimensional roots
• Discrete random numbers, Discrete random numbers, Discrete random numbers, Discrete random numbers
• Discrete random numbers, preprocessing
• divided differences, polynomials
• division by zero, IEEE exceptions
• dollar sign `\$`, shell prompt
• DOT, Level-1 BLAS
• double factorial
• double precision, IEEE format
• DWT initialization
• DWT, mathematical definition
• DWT, one dimensional
• DWT, see wavelet transforms
• DWT, two dimensional
• ## e

• e, defined as a macro
• E1(x), E2(x), Ei(x)
• eigenvalues and eigenvectors
• elementary functions
• elementary operations
• elliptic functions (Jacobi)
• elliptic integrals
• energy function
• energy, units of
• erf(x)
• erfc(x)
• Erlang distribution
• error codes
• error codes, reserved
• error function
• Error handlers
• error handling
• error handling macros
• Errors
• estimated standard deviation
• estimated variance
• Eta Function
• euclidean distance function, hypot, euclidean distance function, hypot
• Euler's constant, defined as a macro
• evaluation of polynomials
• evaluation of polynomials, in divided difference form
• examples, conventions used in
• exceptions, C++
• exceptions, floating point
• exceptions, IEEE arithmetic
• exchanging permutation elements
• exp
• expm1
• exponent, IEEE format
• Exponential distribution
• exponential function
• exponential integrals
• Exponential power distribution
• exponential, difference from 1 computed accurately
• exponentiation of complex number
• `extern inline`
• ## f

• F-distribution
• factorial, factorial
• factorization of matrices
• false position algorithm for finding roots
• Fast Fourier Transforms, see FFT
• Fehlberg method, differential equations
• Fermi-Dirac function
• FFT
• FFT mathematical definition
• FFT of complex data, mixed-radix algorithm
• FFT of complex data, radix-2 algorithm
• FFT of real data
• FFT of real data, mixed-radix algorithm
• FFT of real data, radix-2 algorithm
• FFT, complex data
• finding minima
• finding roots
• finding zeros
• fits, multi-parameter linear
• fitting
• fitting, using Chebyshev polynomials
• Fj(x), Fermi-Dirac integral
• Fj(x,b), incomplete Fermi-Dirac integral
• flat distribution
• Fletcher-Reeves conjugate gradient algorithm, minimization
• floating point exceptions
• floating point numbers, approximate comparison
• floating point registers
• force and energy, units of
• Fortran range checking, equivalent in gcc
• Four-tap Generalized Feedback Shift Register
• Fourier integrals, numerical
• Fourier Transforms, see FFT
• Fractional Order Bessel Functions
• free software, explanation of
• frexp
• functions, numerical differentiation
• fundamental constants
• ## g

• Gamma distribution
• gamma functions
• Gaussian distribution
• Gaussian distribution, bivariate
• Gaussian Tail distribution
• gcc extensions, range-checking
• gcc warning options
• gdb
• Gegenbauer functions
• GEMM, Level-3 BLAS
• GEMV, Level-2 BLAS
• general polynomial equations, solving
• generalized eigensystems
• generalized hermitian definite eigensystems
• generalized symmetric eigensystems
• Geometric random variates, Geometric random variates
• GER, Level-2 BLAS
• GERC, Level-2 BLAS
• GERU, Level-2 BLAS
• Givens Rotation, BLAS
• Givens Rotation, Modified, BLAS
• golden section algorithm for finding minima
• GSL_C99_INLINE
• `GSL_RNG_SEED`
• gsl_sf_result
• gsl_sf_result_e10
• Gumbel distribution (Type 1)
• Gumbel distribution (Type 2)
• ## h

• Haar wavelets
• Hankel transforms, discrete
• HAVE_INLINE
• hazard function, normal distribution
• HBOOK
• heapsort
• HEMM, Level-3 BLAS
• HEMV, Level-2 BLAS
• HER, Level-2 BLAS
• HER2, Level-2 BLAS
• HER2K, Level-3 BLAS
• HERK, Level-3 BLAS
• hermitian matrix, complex, eigensystem
• Hessenberg decomposition
• Hessenberg triangular decomposition
• histogram statistics
• histogram, from ntuple
• histograms
• histograms, random sampling from
• Householder linear solver
• Householder matrix
• Householder transformation
• Hurwitz Zeta Function
• HYBRID algorithm, unscaled without derivatives
• HYBRID algorithms for nonlinear systems
• HYBRIDJ algorithm
• HYBRIDS algorithm, scaled without derivatives
• HYBRIDSJ algorithm
• hydrogen atom
• hyperbolic cosine, inverse
• hyperbolic functions, complex numbers
• hyperbolic integrals
• hyperbolic sine, inverse
• hyperbolic space
• hyperbolic tangent, inverse
• hypergeometric functions
• hypergeometric random variates
• hypot
• hypot function, special functions
• ## i

• i(x), Bessel Functions
• I(x), Bessel Functions
• identity matrix
• identity permutation
• IEEE exceptions
• IEEE floating point
• IEEE format for floating point numbers
• IEEE infinity, defined as a macro
• IEEE NaN, defined as a macro
• illumination, units of
• imperial units
• Implicit Euler method
• Implicit Runge-Kutta method
• importance sampling, VEGAS
• incomplete Beta function, normalized
• incomplete Fermi-Dirac integral
• incomplete Gamma function
• indirect sorting
• indirect sorting, of vector elements
• infinity, defined as a macro
• infinity, IEEE format
• info-gsl mailing list
• initial value problems, differential equations
• initializing matrices
• initializing vectors
• inline functions
• integer powers
• integrals, exponential
• interpolation
• interpolation, using Chebyshev polynomials
• inverse complex trigonometric functions
• inverse cumulative distribution functions
• inverse hyperbolic cosine
• inverse hyperbolic functions, complex numbers
• inverse hyperbolic sine
• inverse hyperbolic tangent
• inverse of a matrix, by LU decomposition
• inverting a permutation
• Irregular Cylindrical Bessel Functions
• Irregular Modified Bessel Functions, Fractional Order
• Irregular Modified Cylindrical Bessel Functions
• Irregular Modified Spherical Bessel Functions
• Irregular Spherical Bessel Functions
• iterating through combinations
• iterating through multisets
• iterating through permutations
• iterative refinement of solutions in linear systems
• ## j

• j(x), Bessel Functions
• J(x), Bessel Functions
• Jacobi elliptic functions
• Jacobi orthogonalization
• Jacobian matrix, fitting
• Jacobian matrix, ODEs
• Jacobian matrix, root finding
• ## k

• k(x), Bessel Functions
• K(x), Bessel Functions
• knots, basis splines
• kurtosis
• ## l

• Laguerre functions
• Lambert function
• Landau distribution
• LAPACK
• Laplace distribution
• LD_LIBRARY_PATH
• ldexp
• least squares fit
• least squares fitting, nonlinear
• least squares troubleshooting
• least squares, covariance of best-fit parameters
• Legendre forms of elliptic integrals
• Legendre functions
• Legendre polynomials
• length, computed accurately using hypot, length, computed accurately using hypot
• Levenberg-Marquardt algorithms
• Levin u-transform
• Levy distribution
• Levy distribution, skew
• libraries, shared
• light, units of
• linear algebra
• linear algebra, BLAS
• linear interpolation
• linear regression
• linear systems, refinement of solutions
• linear systems, solution of
• LMDER algorithm
• log1p
• logarithm and related functions
• logarithm of Beta function
• logarithm of combinatorial factor C(m,n)
• logarithm of complex number
• logarithm of cosh function, special functions
• logarithm of double factorial
• logarithm of factorial
• logarithm of Gamma function
• logarithm of Pochhammer symbol
• logarithm of sinh function, special functions
• logarithm of the determinant of a matrix
• logarithm, computed accurately near 1
• Logarithmic random variates
• Logistic distribution
• Lognormal distribution
• long double
• low discrepancy sequences
• Low-level CBLAS
• LU decomposition
• ## m

• macros for mathematical constants
• magnitude of complex number
• mailing list archives
• mailing list for GSL announcements
• mailing list, bug-gsl
• mantissa, IEEE format
• mass, units of
• mathematical constants, defined as macros
• mathematical functions, elementary
• Mathieu Function Characteristic Values
• Mathieu functions
• matrices, matrices
• matrices, initializing
• matrices, range-checking
• matrix determinant
• matrix diagonal
• matrix factorization
• matrix inverse
• matrix square root, Cholesky decomposition
• matrix subdiagonal
• matrix superdiagonal
• matrix, constant
• matrix, identity
• matrix, operations
• matrix, zero
• max
• maximal phase, Daubechies wavelets
• maximization, see minimization
• maximum of two numbers
• maximum value, from histogram
• mean
• mean value, from histogram
• Mills' ratio, inverse
• min
• minimization, BFGS algorithm
• minimization, caveats
• minimization, multidimensional
• minimization, one-dimensional
• minimization, overview
• minimization, Polak-Ribiere algorithm
• minimization, providing a function to minimize
• minimization, simplex algorithm
• minimization, steepest descent algorithm
• minimization, stopping parameters
• minimum finding, Brent's method
• minimum finding, golden section algorithm
• minimum of two numbers
• minimum value, from histogram
• MINPACK, minimization algorithms, MINPACK, minimization algorithms
• MISCFUN
• MISER monte carlo integration
• Modified Bessel Functions, Fractional Order
• Modified Cylindrical Bessel Functions
• Modified Givens Rotation, BLAS
• Modified Newton's method for nonlinear systems
• Modified Spherical Bessel Functions
• Monte Carlo integration
• MRG, multiple recursive random number generator
• MT19937 random number generator
• multi-parameter regression
• multidimensional integration
• multidimensional root finding, Broyden algorithm
• multidimensional root finding, overview
• multidimensional root finding, providing a function to solve
• Multimin, caveats
• Multinomial distribution
• multiplication
• multisets
• multistep methods, ODEs
• ## n

• N-dimensional random direction vector
• NaN, defined as a macro
• nautical units
• Negative Binomial distribution, random variates
• Nelder-Mead simplex algorithm for minimization
• Newton algorithm, discrete
• Newton algorithm, globally convergent
• Newton's method for finding roots
• Newton's method for systems of nonlinear equations
• Niederreiter sequence
• NIST Statistical Reference Datasets
• non-normalized incomplete Gamma function
• nonlinear equation, solutions of
• nonlinear fitting, stopping parameters
• nonlinear functions, minimization
• nonlinear least squares fitting
• nonlinear least squares fitting, overview
• nonlinear systems of equations, solution of
• nonsymmetric matrix, real, eigensystem
• Nordsieck form
• normalized form, IEEE format
• normalized incomplete Beta function
• Not-a-number, defined as a macro
• NRM2, Level-1 BLAS
• ntuples
• nuclear physics, constants
• numerical constants, defined as macros
• numerical derivatives
• ## o

• obtaining GSL
• ODEs, initial value problems
• optimization, combinatorial
• optimization, see minimization
• optimized functions, alternatives
• ordering, matrix elements
• ordinary differential equations, initial value problem
• oscillatory functions, numerical integration of
• overflow, IEEE exceptions
• ## p

• Pareto distribution
• PAW
• permutations
• physical constants
• physical dimension, matrices
• pi, defined as a macro
• plain Monte Carlo
• Pochhammer symbol
• Poisson random numbers
• Polak-Ribiere algorithm, minimization
• polar form of complex numbers
• polar to rectangular conversion
• polygamma functions
• polynomial evaluation
• polynomial interpolation
• polynomials, roots of
• power function
• power of complex number
• power, units of
• precision, IEEE arithmetic
• predictor-corrector method, ODEs
• prefixes
• pressure, units of
• Prince-Dormand, Runge-Kutta method
• printers units
• probability distribution, from histogram
• probability distributions, from histograms
• projection of ntuples
• psi function
• ## q

• QR decomposition
• QR decomposition with column pivoting
• quantile functions
• quasi-random sequences
• ## r

• R250 shift-register random number generator
• Racah coefficients
• Radix-2 FFT for real data
• rand, BSD random number generator
• rand48 random number generator
• random number distributions
• random number generators
• random sampling from histograms
• RANDU random number generator
• RANF random number generator
• range
• range-checking for matrices
• range-checking for vectors
• RANLUX random number generator
• RANLXD random number generator
• RANLXS random number generator
• RANMAR random number generator, RANMAR random number generator
• Rayleigh distribution
• Rayleigh Tail distribution
• real nonsymmetric matrix, eigensystem
• real symmetric matrix, eigensystem
• Reciprocal Gamma function
• rectangular to polar conversion
• recursive stratified sampling, MISER
• reduction of angular variables
• refinement of solutions in linear systems
• regression, least squares
• Regular Bessel Functions, Fractional Order
• Regular Bessel Functions, Zeros of
• Regular Cylindrical Bessel Functions
• Regular Modified Bessel Functions, Fractional Order
• Regular Modified Cylindrical Bessel Functions
• Regular Modified Spherical Bessel Functions
• Regular Spherical Bessel Functions
• Regulated Gamma function
• relative Pochhammer symbol
• reporting bugs in GSL
• representations of complex numbers
• resampling from histograms
• residual, in nonlinear systems of equations, residual, in nonlinear systems of equations
• reversing a permutation
• Riemann Zeta Function
• RK2, Runge-Kutta method
• RK4, Runge-Kutta method
• RKF45, Runge-Kutta-Fehlberg method
• robust regression
• root finding
• root finding, bisection algorithm
• root finding, Brent's method
• root finding, caveats
• root finding, false position algorithm
• root finding, initial guess
• root finding, Newton's method
• root finding, overview
• root finding, providing a function to solve
• root finding, search bounds
• root finding, secant method
• root finding, Steffenson's method
• root finding, stopping parameters, root finding, stopping parameters
• roots
• ROTG, Level-1 BLAS
• rounding mode
• Runge-Kutta Cash-Karp method
• Runge-Kutta methods, ordinary differential equations
• Runge-Kutta Prince-Dormand method
• ## s

• safe comparison of floating point numbers
• safeguarded step-length algorithm
• sampling from histograms, sampling from histograms
• SAXPY, Level-1 BLAS
• SCAL, Level-1 BLAS
• schedule, cooling
• se(q,x), Mathieu function
• secant method for finding roots
• selection function, ntuples
• series, acceleration
• shared libraries
• shell prompt
• Shi(x)
• shift-register random number generator
• Si(x)
• sign bit, IEEE format
• sign of the determinant of a matrix
• simplex algorithm, minimization
• simulated annealing
• sin, of complex number
• sine function, special functions
• single precision, IEEE format
• singular functions, numerical integration of
• singular points, specifying positions in quadrature
• singular value decomposition
• Skew Levy distribution
• skewness
• slope, see numerical derivative
• Sobol sequence
• solution of linear system by Householder transformations
• solution of linear systems, Ax=b
• solving a nonlinear equation
• solving nonlinear systems of equations
• sorting
• sorting eigenvalues and eigenvectors
• sorting vector elements
• source code, reuse in applications
• special functions
• Spherical Bessel Functions
• spherical harmonics
• spherical random variates, 2D
• spherical random variates, 3D
• spherical random variates, N-dimensional
• spline
• splines, basis
• square root of a matrix, Cholesky decomposition
• square root of complex number
• standard deviation
• standard deviation, from histogram
• standards conformance, ANSI C
• Statistical Reference Datasets (StRD)
• statistics
• statistics, from histogram
• steepest descent algorithm, minimization
• Steffenson's method for finding roots
• stratified sampling in Monte Carlo integration
• stride, of vector index
• Student t-distribution
• subdiagonal, of a matrix
• summation, acceleration
• superdiagonal, matrix
• SVD
• SWAP, Level-1 BLAS
• swapping permutation elements
• SYMM, Level-3 BLAS
• symmetric matrix, real, eigensystem
• SYMV, Level-2 BLAS
• synchrotron functions
• SYR, Level-2 BLAS
• SYR2, Level-2 BLAS
• SYR2K, Level-3 BLAS
• SYRK, Level-3 BLAS
• systems of equations, nonlinear
• ## t

• t-distribution
• t-test
• tangent of complex number
• Tausworthe random number generator
• Taylor coefficients, computation of
• testing combination for validity
• testing multiset for validity
• testing permutation for validity
• thermal energy, units of
• time units
• trailing dimension, matrices
• transformation, Householder
• transforms, Hankel
• transforms, wavelet
• transport functions
• traveling salesman problem
• tridiagonal decomposition, tridiagonal decomposition
• tridiagonal systems
• trigonometric functions
• trigonometric functions of complex numbers
• trigonometric integrals
• TRMM, Level-3 BLAS
• TRMV, Level-2 BLAS
• TRSM, Level-3 BLAS
• TRSV, Level-2 BLAS
• TSP
• TT800 random number generator
• two dimensional Gaussian distribution
• two dimensional histograms
• two-sided exponential distribution
• Type 1 Gumbel distribution, random variates
• Type 2 Gumbel distribution
• ## u

• u-transform for series
• underflow, IEEE exceptions
• uniform distribution
• units, conversion of
• units, imperial
• Unix random number generators, rand
• Unix random number generators, rand48
• unnormalized incomplete Gamma function
• unweighted linear fits
• usage, compiling application programs
• ## v

• value function, ntuples
• Van der Pol oscillator, example
• variance
• variance, from histogram
• variance-covariance matrix, linear fits
• VAX random number generator
• vector, operations
• vector, sorting elements of
• vectors, vectors
• vectors, initializing
• vectors, range-checking
• VEGAS Monte Carlo integration
• viscosity, units of
• volume units
• ## w

• W function
• warning options
• warranty (none)
• wavelet transforms
• website, developer information
• Weibull distribution
• weight, units of
• weighted linear fits
• Wigner coefficients
• ## y

• Y(x), Bessel Functions
• y(x), Bessel Functions
• ## z

• zero finding
• zero matrix
• zero, IEEE format
• Zeros of Regular Bessel Functions
• Zeta functions
• Ziggurat method