Multiple precision evaluation of the Airy Ai function with reduced cancellation

TL;DR

This paper presents a new method for accurately evaluating the Airy Ai function for positive x by decomposing it into well-conditioned functions and using Miller's algorithm, enabling certified arbitrary precision calculations.

Contribution

It introduces a novel decomposition of Ai(x) into functions with nonnegative Taylor series, improving evaluation stability and enabling certified high-precision computation.

Findings

The method achieves well-conditioned series for Ai(x) evaluation.

Miller's algorithm effectively handles the ill-conditioned recurrence.

The implementation provides certified arbitrary precision results.

Abstract

The series expansion at the origin of the Airy function Ai(x) is alternating and hence problematic to evaluate for x > 0 due to cancellation. Based on a method recently proposed by Gawronski, M\"uller, and Reinhard, we exhibit two functions F and G, both with nonnegative Taylor expansions at the origin, such that Ai(x) = G(x)/F(x). The sums are now well-conditioned, but the Taylor coefficients of G turn out to obey an ill-conditioned three-term recurrence. We use the classical Miller algorithm to overcome this issue. We bound all errors and our implementation allows an arbitrary and certified accuracy, that can be used, e.g., for providing correct rounding in arbitrary precision.