Fresnel Integral Computation Techniques

TL;DR

This paper presents a comprehensive method for accurately and efficiently computing Fresnel Integrals across different argument ranges using Taylor series, trapezoid rule, and asymptotic expansion, with error estimates and parameter selection guidance.

Contribution

It introduces a systematic approach for choosing computation parameters for arbitrary precision in Fresnel Integral evaluation, extending previous methods.

Findings

Accurate computation of Fresnel Integrals across all argument ranges.

Provision of error estimates and parameter selection guidelines.

Numerical verification of the method's effectiveness with double precision.

Abstract

This work is an extension of previous work by Alazah et al. [M. Alazah, S. N. Chandler-Wilde, and S. La Porte, Numerische Mathematik, 128(4):635-661, 2014]. We split the computation of the Fresnel Integrals into 3 cases: a truncated Taylor series, modified trapezoid rule and an asymptotic expansion for small, medium and large arguments respectively. These special functions can be computed accurately and efficiently up to an arbitrary precision. Error estimates are provided and we give a systematic method in choosing the various parameters for a desired precision. We illustrate this method and verify numerically using double precision.