Basic Methods for Computing Special Functions

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.

[1]  J. Rice.  The approximation of functions , 1964 .

[2]  B. C. Carlson.  Numerical computation of real or complex elliptic integrals , 2005, Numerical Algorithms.

[3]  Robert C. Forrey,et al.  Computing the Hypergeometric Function , 1997 .

[4]  G. A. Baker.  Essentials of Padé approximants , 1975 .

[5]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[6]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[7]  John P. Boyd,et al.  The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series , 1999 .

[8]  P. Wynn,et al.  On a device for computing the _{}(_{}) tranformation , 1956 .

[9]  N. Temme.  Numerical algorithms for uniform Airy-type asymptotic expansions , 2004, Numerical Algorithms.

[10]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[11]  M. Abramowitz,et al.  Mathematical functions and their approximations , 1975 .

[12]  Nico M. Temme,et al.  Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions , 2002, TOMS.

[13]  L. Trefethen,et al.  Talbot quadratures and rational approximations , 2006 .

[14]  Louis Baker,et al.  C mathematical function handbook , 1991 .

[15]  B. C. Carlson.  Computing elliptic integrals by duplication , 1979 .

[16]  Claude Brezinski,et al.  History of continued fractions and Pade approximants , 1990, Springer series in computational mathematics.

[17]  W. G. Bickley,et al.  British Association Mathematical Tables, Vol. X. Bessel Functions, Part II: Functions of Positive Integer Order , 1953 .

[18]  採編典藏組.  Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[19]  Nico M. Temme,et al.  Identifying minimal and dominant solutions for Kummer recursions , 2008, Math. Comput..

[20]  Almerico Murli,et al.  Algorithm 682: Talbot's method of the Laplace inversion problems , 1990, TOMS.

[21]  F. W. J. Olver,et al.  On the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations , 1998, SIAM Rev..

[22]  Nico M. Temme,et al.  Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions , 2002, TOMS.

[23]  W. Gautschi.  Computational Aspects of Three-Term Recurrence Relations , 1967 .

[24]  Javier Segura.  Reliable Computation of the Zeros of Solutions of Second Order Linear ODEs Using a Fourth Order Method , 2010, SIAM J. Numer. Anal..

[25]  G. Maino,et al.  Computation of parabolic cylinder functions by means of a tricomi expansion , 1981 .

[26]  C. Noble.  Evaluation of negative energy Coulomb (Whittaker) functions , 2004 .

[27]  Amparo Gil,et al.  DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics☆ , 2001 .

[28]  Claude Brezinski,et al.  Extrapolation methods - theory and practice , 1993, Studies in computational mathematics.

[29]  Bruce R. Fabijonas,et al.  Algorithm 838: Airy Functions , 2004, TOMS.

[30]  Nico M. Temme,et al.  Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments , 2004, TOMS.

[31]  Stephan Fritzsche,et al.  Maple procedures for the coupling of angular momenta. IV.Spherical harmonics ? ? This program can b , 2001 .

[32]  David M. Smith.  Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions , 2001, TOMS.

[33]  John F. Hart,et al.  Computer Approximations , 1978 .

[34]  P. Clarkson,et al.  The second Painlevé equation, its hierarchy and associated special polynomials , 2003 .

[35]  P. F. Córdoba,et al.  A code to evaluate modified bessel functions based on thecontinued fraction method , 1997 .

[36]  Ernst Joachim Weniger,et al.  Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series , 1989 .

[37]  Nico M. Temme,et al.  Numerical methods for special functions , 2007 .

[38]  M. Seaton.  Coulomb functions for attractive and repulsive potentials and for positive and negative energies , 2002 .

[39]  Amparo Gil,et al.  Evaluation of legendre functions of argument greater than one , 1997 .

[40]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[41]  Padma Raghavan,et al.  Parallel Processing for Scientific Computing , 2006, Software, Environments, Tools.

[42]  Masao Kodama,et al.  Algorithm 877: A Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument , 2008, TOMS.

[43]  Ernst Joachim Weniger,et al.  On the summation of some divergent hypergeometric series and related perturbation expansions , 1990 .

[44]  Ernst Joachim Weniger,et al.  The summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic, sextic, and octic anharmonic oscillators using nonlinear sequence transformations , 1993 .

[45]  Nico M. Temme,et al.  Algorithm 831: Modified Bessel functions of imaginary order and positive argument , 2004, TOMS.

[46]  W. Read,et al.  A CORRECTION TO A HIGHLY ACCURATE VOIGT FUNCTION ALGORITHM , 2003 .

[47]  J. Rice,et al.  The Approximation of Functions, Vol. 1: Linear Theory , 1965 .

[48]  C. W. Clenshaw.  The numerical solution of linear differential equations in Chebyshev series , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[49]  Nico M. Temme,et al.  Numerically satisfactory solutions of Kummer recurrence relations , 2008, Numerische Mathematik.

[50]  G. Meinardus.  Approximation of Functions: Theory and Numerical Methods , 1967 .

[51]  David Levin,et al.  Development of non-linear transformations for improving convergence of sequences , 1972 .

[52]  J. H. Johnson,et al.  REMES 2: A FORTRAN program to calculate rational minimax approximations to a given function , 1973 .

[53]  Walter Gautschi,et al.  Mathematics of computation, 1943-1993 : a half-century of computational mathematics : Mathematics of Computation 50th Anniversary Symposium, August 9-13, 1993, Vancouver, British Columbia , 1994 .

[54]  Stephen Lloyd Baluk Moshier,et al.  Methods and programs for mathematical functions , 1989 .

[55]  J. Gard,et al.  Method for Evaluation of Zeros of Bessel Functions , 1973 .

[56]  C. Brezinski,et al.  Extrapolation methods , 1992 .

[57]  William H. Press,et al.  Numerical recipes in C , 2002 .

[58]  Mariarosaria Rizzardi,et al.  A modification of Talbot's method for the simultaneous approximation of several values of the inverse Laplace transform , 1995, TOMS.

[59]  David M. Miller,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[60]  J. L. Schonfelder,et al.  Chebyshev expansions for the error and related functions , 1978 .

[61]  Tobias Huber,et al.  HypExp 2, Expanding hypergeometric functions about half-integer parameters , 2007, Comput. Phys. Commun..

[62]  Fayez A. Alhargan,et al.  Algorithm 804: subroutines for the computation of Mathieu functions of integer orders , 2000, TOMS.

[63]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[64]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[65]  C. W. Clenshaw.  A note on the summation of Chebyshev series , 1955 .

[66]  Nico M. Temme,et al.  Numerically satisfactory solutions of hypergeometric recursions , 2007, Math. Comput..

[67]  Jean Marie Linhart,et al.  Algorithm 885: Computing the Logarithm of the Normal Distribution , 2008, TOMS.

[68]  Asymptotic expansions and converging factors I. General theory and basic converging factors , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[69]  Stan Wagon,et al.  The SIAM 100-Digit Challenge - A study in High-Accuracy Numerical Computing , 2004, The SIAM 100-Digit Challenge.

[70]  N. Temme.  Special Functions: An Introduction to the Classical Functions of Mathematical Physics , 1996 .

[71]  Sanjukta Chatterjee,et al.  A class of new transforms tailored for the hypergeometric series , 2008, Comput. Phys. Commun..

[72]  Lloyd N. Trefethen,et al.  Parabolic and hyperbolic contours for computing the Bromwich integral , 2007, Math. Comput..

[73]  N. M. Temme,et al.  On the numerical evaluation of the modified bessel function of the third kind , 1975 .

[74]  Leon M. Hall,et al.  Special Functions , 1998 .

[75]  D. E. Roberts,et al.  The epsilon algorithm and related topics , 2000 .

[76]  C. W. Clenshaw.  Chebyshev series for mathematical functions , 1962 .

[77]  John R. Airey D.Sc. Sc.D..  LII. The “converging factor” in asymptotic series and the calculation of Bessel, laguerre and other functions , 1937 .

[78]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[79]  Lloyd N. Trefethen,et al.  Computing the Gamma Function Using Contour Integrals and Rational Approximations , 2007, SIAM J. Numer. Anal..

[80]  A V Hershey,et al.  Computation of Special Functions , 1978 .

[81]  Ranjan Bhattacharya,et al.  Rational interpolation using Levin-Weniger transforms , 1997 .

[82]  Amparo Gil,et al.  Evaluation of toroidal harmonics , 2000 .

[83]  Danilo Erricolo.  Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch's algorithm , 2006, TOMS.

[84]  M. V. Stoitsov,et al.  Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions , 2007, Comput. Phys. Commun..

[85]  Ronald Cools,et al.  Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions , 2006, TOMS.

[86]  Nico M. Temme,et al.  Computing Complex Airy Functions by Numerical Quadrature , 2002, Numerical Algorithms.

[87]  L. Wuytack,et al.  Padé Approximation and its Applications , 1979 .

[88]  K. Brueckner,et al.  Advances in theoretical physics , 1965 .

[89]  J. A. C. Weideman,et al.  Optimizing Talbot's Contours for the Inversion of the Laplace Transform , 2006, SIAM J. Numer. Anal..

[90]  N. Michel.  Precise Coulomb wave functions for a wide range of complex l, eta and z , 2007, Comput. Phys. Commun..

[91]  S. Graffi,et al.  Borel summability and indeterminacy of the Stieltjes moment problem: Application to the anharmonic oscillators , 1978 .

[92]  Nico M. Temme,et al.  An algorithm with ALGOL 60 program for the computation of the zeros of ordinary bessel functions and those of their derivatives , 1979 .

[93]  William J. Thompson,et al.  Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners with Programs in Fortran 90 and Mathematica , 1997 .

[94]  Walter Gautschi,et al.  A Computational Procedure for Incomplete Gamma Functions , 1979, TOMS.

[95]  D. E. Amos.  Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order , 1986, TOMS.

[96]  A. Talbot.  The Accurate Numerical Inversion of Laplace Transforms , 1979 .

[97]  Roy G. Gordon,et al.  An algorithm for the evaluation of the complex Airy functions , 1979 .

[98]  F. D. Colavecchiaa,et al.  f 1 : a code to compute Appell ’ s F 1 hypergeometric function ✩ , 2004 .

[99]  Annie A. M. Cuyt,et al.  Handbook of Continued Fractions for Special Functions , 2008 .

[100]  Roy G. Gordon,et al.  A numerical algorithm for the evaluation of Weber parabolic cylinder functions U(a,x), V(a,x), and W(a, ±x)☆ , 1981 .

[101]  Nico M. Temme,et al.  Algorithm 850: Real parabolic cylinder functions U(a, x), V(a, x) , 2006, TOMS.

[102]  A. R. DiDonato,et al.  New Formulas for Computing Incomplete Elliptic Integrals of the First and Second Kind , 1959, JACM.

[103]  Daniel W. Lozier,et al.  Airy and Bessel Functions by Parallel Integration of ODEs , 1993, PPSC.

[104]  Herbert H. H. Homeier.  Scalar Levin-type sequence transformations , 2000 .

[105]  Ernst Joachim Weniger,et al.  rational approximations for the modified Bessel function of the second kind , 1990 .

[106]  Amparo Gil,et al.  Computing the Zeros and Turning Points of Solutions of Second Order Homogeneous Linear ODEs , 2003, SIAM J. Numer. Anal..

[107]  Claude Brezinski,et al.  Convergence acceleration during the 20th century , 2000 .

[108]  M. Seaton.  numer, a code for Numerov integrations of Coulomb functions ? ? This program can be downloaded from , 2002 .

[109]  F. D. Colavecchia,et al.  f1: a code to compute Appell's F1 hypergeometric function , 2004 .

[110]  Nico M. Temme,et al.  Fast and accurate computation of the Weber parabolic cylinder function W(a, x) , 2011 .

[111]  W. J. Cody,et al.  A Survey of Practical Rational and Polynomial Approximation of Functions , 1970 .

[112]  Walter Gautschi,et al.  NUMERICAL EVALUATION OF SPECIAL FUNCTIONS , 2001 .

[113]  A. Gil,et al.  Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments , 1998 .

[114]  P. Wynn,et al.  Upon systems of recursions which obtain among the quotients of the Padé table , 1966 .

[115]  Walter Gautschi,et al.  Computation of Bessel and Airy Functions and of Related Gaussian Quadrature Formulae , 2002 .

[116]  Fayez A. Alhargan.  Algorithm 855: Subroutines for the computation of Mathieu characteristic numbers and their general orders , 2006, TOMS.

[117]  M. J. D. Powell,et al.  On the Maximum Errors of Polynomial Approximations Defined by Interpolation and by Least Squares Criteria , 1967, Comput. J..

[118]  A. Gray.  Bessel Functions , 1899, Nature.

[119]  Nico M. Temme,et al.  Computing the Conical Function Pµ-1/2+iTau(x) , 2009, SIAM J. Sci. Comput..

[120]  Amparo Gil,et al.  Computing Toroidal Functions for Wide Ranges of the Parameters , 2000, Journal of Computational Physics.

[121]  Allan MacLeod.  An instability problem in Chebyshev expansions for special functions , 1993, SGNM.

[122]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[123]  Amparo Gil,et al.  A code to evaluate prolate and oblate spheroidal harmonics , 1998 .

[124]  G. A. Baker,et al.  THE THEORY AND APPLICATION OF THE PADE APPROXIMANT METHOD , 1964 .

[125]  Daniel W. Lozier,et al.  Computation of complex Airy functions and their zeros using asymptotics and the differential equation , 2004, TOMS.

[126]  J. D. Talman.  NumSBT: A subroutine for calculating spherical Bessel transforms numerically , 2009, Comput. Phys. Commun..

[127]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[128]  Nico M. Temme,et al.  The ABC of hyper recursions , 2004 .

[129]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

[130]  Yasuhiko Ikebe,et al.  The zeros of regular Coulomb wave functions and of their derivatives , 1975 .

[131]  Javier Segura,et al.  The Zeros of Special Functions from a Fixed Point Method , 2002, SIAM J. Numer. Anal..

[132]  Nico M. Temme,et al.  Computing the real parabolic cylinder functions U(a, x), V(a, x) , 2006, TOMS.

[133]  Nico M. Temme,et al.  On nonoscillating integrals for computing inhomogeneous Airy functions , 2001, Math. Comput..

[134]  T. Stieltjes,et al.  Recherches sur quelques séries semi-convergentes , 1886 .

[135]  C. Loan.  Computational Frameworks for the Fast Fourier Transform , 1992 .

[136]  KENDALL E. ATKINSON,et al.  Algorithm 876: Solving Fredholm Integral Equations of the Second Kind in Matlab , 2008, TOMS.