Inventory of Mathematical Functions and Algorithms |
| LCG Project | LCG Applications Area | Cernlib writeup | GSL contents | Abramowitz and Stegun | MathLib Project | Project Portal | $Date: 2007/09/21 15:01:05 $ |
| Functions and Polynomials | Numerical Methods | Random Numbers and Distributions | Others |
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Special Functions |
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| Routines for evaluating Special functions | |||
| Bessel Functions of various types | |||
| * | Regular cylindrical functions | Bessel J functions of various orders | ROOT, GSL, Cernlib |
| * | Irregular cylindrical functions | Bessel Y functions of various orders | ROOT, GSL, Cernlib |
| * | Regular modified cylindrical | Bessel I functions of various orders | ROOT, GSL, Cernlib |
| * | Irregular modified cylindrical | Bessel K functions of various orders | ROOT, GSL, Cernlib |
| * | Regular spherical functions | Bessel j functions of various orders | ROOT, GSL, Cernlib |
| * | Irregular spherical functions | Bessel y functions of various orders | ROOT, GSL, Cernlib |
| Examples and tutorials | |||
| * | Clausen function | Clausen integral function | GSL, Cernlib |
| * | Coulomb Wave Function | Wave functions for bound states and scattering solutions | GSL, Cernlib |
| * | Dawson's integral function | Dawson integral | GSL, Cernlib |
| * | Dilogarithm function | Dilogarithms for real arguments | ROOT, GSL, Cernlib |
| Dilogarithms for complex arguments | GSL, Cernlib | ||
| * | Complete Elliptic integrals | Legendre form of the various types of complete Elliptic integrals | ROOT, GSL, Cernlib |
| * | Uncomplete Elliptic integrals | Carlson and Legendre form of uncomplete Elliptic integrals | ROOT, GSL, Cernlib ( 2 ) |
| * | Error functions | Error function (ERFC) and complementary | ROOT (MathCore, TMath), GSL, Cernlib |
| * | Exponential integrals | Various type of exponential integrals | ROOT, GSL, Cernlib |
| * | Hyperbolic integrals | Sinh and Cosh integrals | GSL |
| * | Trigonometric integrals | Sin and Cos integrals | GSL, Cernlib |
| * | Arctangent integral | Computation of Arctangent integral | GSL, Cernlib |
| * | Fermi Dirac function | Complete and incomplete Fermi-Dirac functions | GSL, Cernlib |
| * | Gamma function | Gamma and related functions | ROOT (MathCore, TMath), GSL, Cernlib( 2, 3) |
| * | Hypergeometric functions | Confluent hypergeometric function |
ROOT, GSL |
| Gauss hypergeometric function | ROOT, GSL | ||
| * | Laguerre functions | Laguerre polynomials of various orders | ROOT, GSL |
| Associate Laguerre polynomials of various orders | ROOT, GSL | ||
| * | Legendre functions | Legendre polynomials, functions and spherical harmonics | ROOT, GSL, Cernlib |
| Associate Legendre polynomials | ROOT, GSL,Cernlib | ||
| * | PSI function | Digamma and polygamma functions | GSL, Cernlib |
| * | Zeta functions (Riemann) | Riemann Zeta function | ROOT, GSL, Cernlib |
| * | Struve functions | Struve funcitons of order Zero and One | ROOT, Cernlib |
| Examples and tutorials | |||
Polynomials | |||
| Algorithms for evaluating and solving polynomials | |||
| * | Polynomial evaluation | Evaluate polynommial of degree n | ROOT, GSL |
| * | Quadratic equations | Find roots (real and complex) of quadratic equation | ROOT, GSL |
| * | Cubic equation | Find roots (real and complex) of cubic equation | ROOT, GSL, Cernlib |
| * | Quartic equations | Find roots (real and complex) of quartic equation | ROOT, Cernlib |
| * | General Polynomial equations | Numerical method fot finding roots of higher order polynomial | ROOT, GSL, Cernlib |
Numerical Integration | |||
| Algorithms for performing numerical integration of a function in one dimension | |||
| * | Non-adaptive Gauss-Kronrod | Simple integration method for smooth functions. | ROOT, GSL |
| * | Adaptive Gauss integration | Gauss-Kronrod adaptive integration agorithms for general functions and for special cases (singularities, infinite intervals, etc..) | ROOT,GSL, Cernlib |
| * | Adaptive integration for integrands with weight functions | Algorithms based on the Clenshaw-Curtis rule vor various cases | GSL, Cernlib |
| * | Multidimensional integration | Monte Carlo algorithms for multi dimensional integrations | ROOT, GSL, Cern ( 2) |
| Examples and tutorials | |||
Numerical differentiation | |||
| Numerical algorithms for calculating derivatives of functions | |||
| * | Numerical derivatives | Compute numerical derivatives by finite differencing | ROOT, GSL, Cernlib |
Linear Algebra | |||
| * | Vector and Matrices operations (BLAS) | Vector operation (BLAS Level 1), Matrix vector operation (BLAS Level
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ROOT(SMatrix, TMatrix), GSL, CBLAS, CLHEP matrix |
| * | Linear Algebra problems | Functions for solving systems of linear equations | |
Random Number generation | |||
| Various random number generators | |||
| * | Random number generators |
Various generators of uniform distributed random numbers, either as integers or double precision floating point numbers | ROOT (TRandom, MathMore), GSL, CLHEP |
| * | Random number distributions |
Generation of random numbers according to specific distributions | ROOT, CLHEP |
| * | Quasi-Random Sequences | Sequences of numbers covering progressively a d-dimensional space | GSL |
Probability Distribution Functions | |||
Calculation of the probability distribution functions and the corresponding cumulative distributions P and Q, inverse and quantiles. |
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| * | Gaussian distributions | One and 2 variable gaussian random numbers, pdf and cumulative distributions | ROOT(MathCore, TMath), GSL, Cernlib (2) |
| * | Exponential distribution | One dim exponential random numbers, pdf and comulative distributions | ROOT, GSL |
| * | Breit Wigner (Cauchy ) distribution | Cauchy (Lorentz) random numbers, pdf and comulative distributions | ROOT(MathCore, TMath), GSL |
| * | Landau distribution | Landau random numbers, pdf and comulative distributions | ROOT, GSL, Cernlib, |
| * | Gamma distribution | Gamma random numbers, pdf and comulative distributions | ROOT(MathCore, TMath), GSL, Cernlib ( 2) |
| * | Flat (uniform) distribution | Flat random numbers, pdf and comulative distributions | ROOT, GSL |
| * | Lognormal distribution | Lognormal random numbers, pdf and comulative distributions | ROOT(MathCore, TMath), GSL |
| * | Chi2-squared distribution | Chi2 random numbers, pdf and comulative distributions | ROOT(MathCore, TMath), GSL, Cernlib ( 2, 3 ) |
| * | F distribution | F random numbers, pdf and comulative distributions | ROOT, GSL |
| * | Student t-distribution | Student-t random numbers, pdf and comulative distributions | ROOT, GSL, Cernlib |
| * | Poisson distribution | Poisson random numbers, pdf and comulative distributions | ROOT (TMath, MathCore), GSL, Cernlib, |
| * | Binomial distribution | Binomial random numbers, pdf and comulative distributions | ROOT, GSL, Cernlib |
| * | Multinomial distribution | Multinomial random numbers, pdf and comulative distributions | GSL, Cernlib |
| * | Kolgomorov distribution | Only Kolgomorov distribution | ROOT, Cernlib |
| * | Vavilov distribution | Vavilov pdf and its inverse | ROOT, Cernlib ( 2) |
Minimization | |||
| Algorithms for finding the minimum (or maximum) of a function in one or more dimensions | |||
| * | One dim. minimization | minimization algorithms for one dimensional functions | ROOT, GSL, Cernlib |
| * | Multidimensional minimization | various algorithms for unconstrained minimization including the calculation of the covariance matrix of best fit parameters | ROOT(TMinuit, Minuit2), GSL |
Root Finding | |||
| Algorithms for finding the root of a function in one or more dimensions | |||
| * | One dimensional root finders | Algorithms to find root of a function in one dimension | ROOT, GSL, Cernlib |
| * | Multidimensional root finders | Algorithms to find n-roots of a set of n functions, to solve a system of non linear equations | GSL, Cernlib |
Interpolation | |||
| various types of algorithms for performing interpolation | |||
| * | Polynomial interpolation | simple method recommended for small number of points | ROOT, GSL |
| * | Cubic spline interpolation | Cubic spline with natural and periodic boundary conditions | ROOT, GSL |
| * | Akima spline interpolation | Method based on the non-rounded corner algorithm of Wodicka | ROOT, GSL |
Function Approximations | |||
| routines for computing approximations to functions | |||
| * | Chebyshev Approximations | routines for create and calculate Chebyshev series with derivatives and integrals | ROOT, GSL, Cernlib ( 2) |
| * | Trigonometric series | summation of trigonometric series | Cernlib |
| * | Series acceleration | method to accelerate convergence of series using Levin u-transform | GSL |
Fast Fourier Transform | |||
| routines for computing discrete Fast Fourier Transforms (FFT) | |||
| * | Complex data FFT | compute FFT (and its inverse) for complex data sequences | ROOT, GSL, Cernlib ( 2) |
| * | Real data FFT | compute FFT (and its inverse) for real data sequences | ROOT, GSL, Cernlib ( 2) |
Differential Equations | |||
| Algorithms for solving n-dimensional first-order systems of differential equations | |||
| * | Range-Kutta algorithms | various types of Range-Kutta solving methods | GSL, Cernlib ( 2) |
| * | Burlirsch-Stoer method | Implicit Bulirsch-Stoer method. Requires Jacobian | GSL, Cernlib |
-- Magdalena Slawinska - 19 Sep 2007
-- David Gonzalez Maline - 19 Feb 2009