Towards reliable software for the evaluation of a class of special functions

Abstract

Special functions are pervasive in all fields of science. The most well-known application areas are in physics, engineering, chemistry, computer science and statistics. Because of their importance, several books and a large collection of papers have been devoted to the numerical computation of these functions. But up to this date, even environments such as Maple, Mathematica, MATLAB and libraries such as IMSL, CERN and NAG offer no routines for the reliable evaluation of special functions. Here the notion reliable indicates that, together with the function evaluation a guaranteed upper bound on the total error or, equivalently, an enclosure for the exact result, is computed. We point out how limit-periodic continued fraction representations of these functions can be helpful in this respect. The newly developed (and implemented) scalable precision technique is mainly based on the use of sharpened a priori truncation error and round-off error upper bounds for real continued fraction representations of special functions of a real variable. The implementation is reliable in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation. © Springer-Verlag Berlin Heidelberg 2006.