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Numerical Evaluation of Special Functions

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


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.

1. Introduction

2. Mathematical Developments

3. Packages, Libraries and Systems

4. Functions of One Variable

5. Functions of Two or More Variables

6. Testing and Library Construction

7. Future Trends


We are grateful to the following individuals for supplying references and making helpful comments: D. E. Amos, W. L. Anderson, A. R. Barnett, E. Battiste, C. Brezinski, B. C. Carlson, B. Gabutti, P. W. Gaffney, W. Gautschi, K. S. Kölbig, S. D. Leigh, L. C. Maximon, M. A. McClain, B. R. Miller, W. Parke, N. M. Temme, M. Vuorinen.

A Note on the Reference Acronyms

In the references that follow, the acronyms within the identifying square brackets follow the AMS BibTeX scheme. Initial letters pertain to the author(s) or editor(s). These are followed by two digits representing the year of publication; a letter may also be appended to distinguish between publications in the same year, e.g. [ AB87a] , [ AB87b] . In the case of papers or books with more than four authors, initial letters from the names of the first three authors are used, followed by a sign, e.g. [ ADK+84] . It is also important to note that the references are listed according to the alphabetical order of the acronyms and not according to the alphabetical order of the authors' names.


G. Allasia and R. Besenghi, Numerical calculation of incomplete gamma functions by the trapezoidal rule, Numer. Math. 50 (1987), 419--428.

G. Allasia and R. Besenghi, Numerical computation of Tricomi's psi function by the trapezoidal rule, Computing 39 (1987), 271--279.

A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, T. V. Pak, and B. S. Pariiskii, Evaluation of prolate spheroidal function by solving the corresponding differential equations, U.S.S.R. Comput. Math. and Math. Phys. 24 (1984), no. 1, 1--11.

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 13:28:57 EDT 1995