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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]
.
References
- Bak92
-
L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New
York, 1992, includes diskette.
- BEGG91
-
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.
- Bis91
-
A. K. Bisoi, A Maple program to generate orthonormal polynomials,
Comput. Math. Appl. 22 (1991), no. 9, 1--5.
- BR91
-
C. Brezinski and M. Redivo Zaglia, A new presentation of orthogonal
polynomials with applications to their computation, Numer. Algorithms
1 (1991), 207--221.
- Chi92
-
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.
- Coo68
-
B. E. Cooper, Algorithm AS 10. The use of orthogonal polynomials,
Appl. Statist. 17 (1968), 283--287.
- EK92
-
Ö. Egecioglu and Ç. K. Koç, A parallel algorithm
for generating discrete orthogonal polynomials, Parallel Comput. 18
(1992), 649--659.
- FG91
-
B. Fischer and G. H. Golub, On generating polynomials which are orthogonal
over several intervals, Math. Comp. 56 (1991), 711--730.
- FG92
-
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.
- Gau82
-
W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist.
Comput. 3 (1982), 289--317.
- Gau85
-
W. Gautschi, Orthogonal polynomials---constructive theory and
applications, J. Comput. Appl. Math. 12/13 (1985), 61--76.
- Gau90
-
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.
- Gau91
-
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.
- Gau94
-
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.
- LPT80
-
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.
- Luk75
-
Y. L. Luke, Mathematical functions and their approximations, Academic
Press, New York, 1975.
- Öpi87
-
U. Öpik, A program to set up systems of orthogonal polynomials,
Comput. Phys. Comm. 46 (1987), 263--296.
- PA92
-
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.
- Sim64
-
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.
- ULI90
-
Mathematical function library for Microsoft--C, United Laboratories,
Inc., John Wiley & Sons, 1990, includes diskettes. Edition also
exists in Fortran (1989).
- Wit68
-
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.
Abstract:
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: dlozier@nist.gov
Institute for Physical Science and Technology,
University of Maryland,
College Park, MD 20742
E-mail address: olver@bessel.umd.edu
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