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Numerical Evaluation of Special Functions

(paper, pdf) (paper, postscript)

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

Mathematical and Computational Sciences Division
National Institute of Standards and Technology
Gaithersburg, Md 20899-8910

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

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


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.

March 1994
December 2000

1991 Mathematics Subject Classification: Primary 65D20; Secondary 33-00.
The research of the second author has been supported by NSF Grant CCR 89-14933.
The December 2000 revision of this paper is available in hard copy from the authors and on the Web at


The article printed in this report will appear in Walter Gautschi (ed.), Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics, American Mathematical Society, Providence, Rhode Island 02940. This report is intended for limited distribution only until the primary publication appears in print.

May 1994

Additional copies of this report have been made for distribution to Digital Library of Mathematical Functions Project Participants (editors, associate editors, authors, validators, and support staff). No changes have been made in the body of the report. The exact bibliographic reference for the published article is [LO94]. A revision is in preparation and will be provided to DLMF Project Participants when ready. It will include references that have appeared in the literature since 1993.

May 2000

This report has been updated through 1999, resulting in an approximate 15% increase in the number of references and a modest expansion of the classification scheme in §4 and §5. The software packages, libraries and systems, described in §3 and cross-referenced in §4 and §5, were re-examined. Maple and Mathematica (§3.4.3 and §3.4.5) were found to have added considerable support for special functions, and three new libraries (§3.2.1, §3.2.3 and §3.3.3) were included. For the Web version, see

December 2000

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Thursday, Jan 11, 2001