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Numerical Evaluation of Special Functions

D. W. Lozier and F. W. J. Olver

5.10. Orthogonal Polynomials .

See also hypergeometric functions (§ 5.5), Legendre functions (§ 5.8), and Weber parabolic cylinder functions (§ 5.13).

5.10.1. Classical Polynomials (Chebyshev, Hermite, Jacobi, Laguerre, Legendre etc. ), Real Arguments.

Software Packages:

[ LPT80, Fortran] , [ Sim64, Algol] , [ Wit68, Fortran] .

Intermediate Libraries:

[ Bak92] , [ ULI90] .

5.10.2. Classical Polynomials, Complex Arguments.

Interactive Systems:

Maple, Mathematica.

5.10.3. Other Orthogonal Polynomials.

Interactive Systems:

[ EK92] .

Software Packages:

[ Bis91, Maple] , [ Coo68, Fortran] , [ Gau94, Fortran] , [ Opi87, Fortran] .

5.10.4. Articles.

[ BEGG91] , [ BR91] , [ Chi92] , [ FG91] , [ FG92] , [ Gau82] , [ Gau85] , [ Gau90] , [ Gau91a] , [ Luk75] , [ PA92] .


L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.

D. L. Boley, S. Elhay, G. H. Golub, and M. H. Gutknecht, Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights, Numer. Algorithms 1 (1991), 21--43.

A. K. Bisoi, A Maple program to generate orthonormal polynomials, Comput. Math. Appl. 22 (1991), no. 9, 1--5.

C. Brezinski and M. Redivo Zaglia, A new presentation of orthogonal polynomials with applications to their computation, Numer. Algorithms 1 (1991), 207--221.

R. C. Y. Chin, A domain decomposition method for generating orthogonal polynomials for a Gaussian weight on a finite interval, J. Comput. Phys. 99 (1992), 321--336.

B. E. Cooper, Algorithm AS 10. The use of orthogonal polynomials, Appl. Statist. 17 (1968), 283--287.

Ö. Egecioglu and Ç. K. Koç, A parallel algorithm for generating discrete orthogonal polynomials, Parallel Comput. 18 (1992), 649--659.

B. Fischer and G. H. Golub, On generating polynomials which are orthogonal over several intervals, Math. Comp. 56 (1991), 711--730.

B. Fischer and G. H. Golub, How to generate unknown orthogonal polynomials out of known orthogonal polynomials, J. Comput. Appl. Math. 43 (1992), 99--115.

W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), 289--317.

W. Gautschi, Orthogonal polynomials---constructive theory and applications, J. Comput. Appl. Math. 12/13 (1985), 61--76.

W. Gautschi, Computational aspects of orthogonal polynomials, Orthogonal Polynomials: Theory and Practice (P. Nevai, ed.), Kluwer Academic Publishers, Dordrecht, 1990, NATO ASI Series, vol. C294, pp. 181--216.

W. Gautschi, Computational problems and applications of orthogonal polynomials, Orthogonal Polynomials and their Applications (C. Brezinski, L. Gori, and A. Ronveaux, eds.), J. C. Baltzer AG, Scientific Publishing Company, Basel, 1991, pp. 61--71.

W. Gautschi, Algorithm. ORTHPOL : A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software 20 (1994), in press.

M. C. Lorenzini, G. Puleo, and A. Tortorici Macaluso, Un package di aritmetica in multiprecisione ed applicazione al calcolo dei polinomi di Jacobi, Hermite, Laguerre, Legendre, Chebyshev, Atti Accad. Sci. Lett. Arti Palermo Ser. 4, Parte 1 39 (1980), 339--376.

Y. L. Luke, Mathematical functions and their approximations, Academic Press, New York, 1975.

U. Öpik, A program to set up systems of orthogonal polynomials, Comput. Phys. Comm. 46 (1987), 263--296.

C. U. Pabon-Ortiz and M. Artoni, Laguerre polynomials : Novel properties and numerical generation scheme for analysis of a discrete probability distribution, Comput. Phys. Comm. 71 (1992), 215--221.

J. M. S. Simões Pereira, Algorithm 234. Poisson-Charlier polynomials, Comm. ACM 7 (1964), 420, for certification see same journal v. 8 (1965), p. 105.

Mathematical function library for Microsoft--C, United Laboratories, Inc., John Wiley & Sons, 1990, includes diskettes. Edition also exists in Fortran (1989).

B. F. W. Witte, Algorithm 332. Jacobi polynomials, Comm. ACM 11 (1968), 436--437, for remarks see same journal v. 13 (1970), p. 449 and v. 18 (1975), pp. 116--117.


This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 1943--1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI 02940, 1994. The symposium was held at the University of British Columbia August 9--13, 1993, in honor of the fiftieth anniversary of the journal Mathematics of Computation.

The original abstract follows.

Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field.

Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Md 20899

E-mail address:

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742

E-mail address:

The research of the second author has been supported by NSF Grant CCR 89-14933.

1991 Mathematics Subject Classification. Primary 65D20; Secondary 33-00.

Daniel W Lozier
Fri Apr 7 14:21:09 EDT 1995