UNRESTRICTED ALGORITHMS FOR ELEMENTARY AND SPECIAL FUNCTIONS

RICHARD P. BRENT

Abstract

We describe some “unrestricted” algorithms which are useful for the computation of elementary and special functions when the precision required is not known in advance. Several general classes of algorithms are identified and illustrated by examples. Applications of such algorithms are mentioned.

Comments

Only the Abstract is given here. The full paper appeared as [3]. For related work see [1, 2].

Errata

Page 614, equation (10), replace “(x/2” by “(x/2)”. Page 617, equation (39), delete “/j!”.

References

[1] R. P. Brent, “Multiple-precision zero-finding methods and the complexity of elementary function evaluation”, in Analytic Computational Complexity (edited by J. F. Traub), Academic Press, New York, 1975, 151–176. MR 52#15938, 54#11843; Zbl 342.65031. rpb028.

[2] R. P. Brent “Fast multiple-precision evaluation of elementary functions”, J. ACM 23 (1976), 242–251. MR 52#16111, Zbl 324.65018. rpb034.

[3] R. P. Brent, “Unrestricted algorithms for elementary and special functions”, invited paper in Information Processing 80 (edited by S. H. Lavington), North-Holland, Amsterdam, 1980, 613–619. CR 22#38728, MR 81i:68009. Also appeared as Report TR-CS-79-13, Department of Computer Science, ANU (November 1979), 20 pp. Retyped in L^{A}TEX, Oxford, October 1999. rpb052.

Department of Computer Science, Australian National University, Canberra

1991 Mathematics Subject Classification. Primary 65D20; Secondary 41A25, 41A99, 65D15, 65B15, 65G99, 68C25.

Key words and phrases. Unrestricted algorithm, multiple precision, high precision, arbitrary precision, elementary function, special function, asymptotic expansion, continued fraction, Euler-Maclaurin sum formula, Bessel function, exponential integral.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1980, IFIP..

Comments \begin{matrix}c \\ ◯ \end{matrix} 1999, R. P. Brent.

rpb052a typeset using AMS-L^{A}TEX.