Unrestricted algorithms for elementary and special functions

52. R. P. Brent, Unrestricted algorithms for elementary and special functions, in Information Processing 80 (edited by S. H. Lavington), North-Holland, Amsterdam, 1980, 613-619. CR 22#38728, MR 81i:68009. Retyped with minor corrections 1999. arXiv:1004.3621v1

Abstract: dvi (2K), pdf (75K), ps (25K).

Paper: dvi (23K) pdf (176K) ps (144K).


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. The topics include: power series methods, use of halving identities, asymptotic expansions, continued fractions, recurrence relations, Newton's method, numerical contour integration, and the arithmetic-geometric mean.


Most of the algorithms discussed are implemented in the MP package. For related work see Brent [28, 34].

Errata for original version

Page 615, remove the absolute value signs in equation (13) and the following paragraph (x should be large and positive here).

Page 616, second half of equation (26): insert a minus sign after the equals sign.

Page 617, equation (39): delete  " / j ! ".

Page 617, equation (42): the assumption "j < k" should be added. Also, the contour C needs to be enlarged slightly.

Page 617, left-hand-side of equation (44): replace "Sj,k" by "S2j,k".

Page 617, ten lines after equation (44): replace "O(jn2)" by "O(j2n)".

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