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 


Special Functions 

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 FermiDirac 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  
*  Nonadaptive GaussKronrod  Simple integration method for smooth functions.  ROOT, GSL 
*  Adaptive Gauss integration  GaussKronrod 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 ClenshawCurtis 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

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 
*  QuasiRandom Sequences  Sequences of numbers covering progressively a ddimensional space  GSL 
Probability Distribution Functions  
Calculation of the probability distribution functions and the corresponding cumulative distributions P and Q, inverse and quantiles. 

*  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 
*  Chi2squared 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 tdistribution  Studentt 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 nroots 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 nonrounded 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 utransform  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 ndimensional firstorder systems of differential equations  
*  RangeKutta algorithms  various types of RangeKutta solving methods  GSL, Cernlib ( 2) 
*  BurlirschStoer method  Implicit BulirschStoer method. Requires Jacobian  GSL, Cernlib 
 Magdalena Slawinska  19 Sep 2007
 David Gonzalez Maline  19 Feb 2009