Unrestricted algorithms for elementary and special functions

TL;DR

This paper introduces various 'unrestricted' algorithms for computing elementary and special functions efficiently when the required precision is unknown beforehand, covering methods like power series, asymptotic expansions, and continued fractions.

Contribution

It presents a comprehensive overview of new algorithms suitable for arbitrary precision computation of special functions, expanding on existing methods with practical implementations.

Findings

Algorithms are effective for high-precision calculations.

Most algorithms are implemented in the MP package.

The methods cover a wide range of computational techniques.

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. 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 (arXiv:1004.3173).