Numerical Evaluation of Special Functions

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

4.7. Zeta Function .

4.7.1. Real Arguments.

Algorithms:

[ CHT71] , [ Luk69b] , [ PB72] .

Software Packages:

[ Mar65, Algol] .

Intermediate Libraries:

[ Bak92] , [ Mos89] .

Interactive Systems:

Maple.

4.7.2. Complex Arguments.

Software Packages:

[ BD80, Fortran] , [ YKK88, Fortran] .

Interactive Systems:

Mathematica.

4.7.3. Articles.

[ AB89] , [ EKK85] , [ Ker80, includes survey] .

References

AB89: G. Allasia and R. Besenghi, Numerical calculation of the Riemann zeta function and generalizations by means of the trapezoidal rule, Numerical and Applied Mathematics, Part 2 (Paris 1988) (C. Brezinski, ed.), IMACS Ann. Comput. Appl. Math., 1.2, Baltzer, Basel, 1989, pp. 467--472.
Bak92: L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.
BD80: A. Bañuelos and R. A. Depine, A program for computing the Riemann zeta function for complex argument, Comput. Phys. Comm. 20 (1980), 441--445.
CHT71: W. J. Cody, K. E. Hillstrom, and H. C. Thacher, Jr., Chebyshev approximations for the Riemann zeta function, Math. Comp. 25 (1971), 537--547.
EKK85: A. Yu. Eremin, I. E. Kaporin, and M. K. Kerimov, The calculation of the Riemann zeta-function in the complex domain, U.S.S.R. Comput. Math. and Math. Phys. 25 (1985), no. 2, 111--119, see also [YKK88].
Ker80: M. K. Kerimov, Methods of computing the Riemann zeta-function and some generalizations of it, U.S.S.R. Comput. Math. and Math. Phys. 20 (1980), no. 6, 212--230.
Luk69: Y. L. Luke, The special functions and their approximations, vol. 2, Academic Press, New York, 1969.
Mar65: B. Markman, Contribution no. 14. The Riemann zeta function, BIT 5 (1965), 138--141.
Mos89: S. L. B. Moshier, Methods and programs for mathematical functions, Ellis Horwood Limited, Chichester, 1989, separate diskette.
PB72: R. Piessens and M. Branders, Chebyshev polynomial expansions of the Riemann zeta function, Math. Comp. 26 (1972), 1022.
YKK88: A. Yu. Yeremin, I. E. Kaporin, and M. K. Kerimov, Computation of the derivatives of the Riemann zeta-function in the complex domain, U.S.S.R. Comput. Math. and Math. Phys. 28 (1988), no. 4, 115--124, see also [EKK85].

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 13:57:20 EDT 1995