Computing Fresnel Integrals via Modified Trapezium Rules

TL;DR

This paper introduces modified trapezium rule methods for computing Fresnel integrals with high accuracy, exponential convergence, and explicit error bounds, suitable for practical implementation across the entire real line.

Contribution

It develops a new approach based on modified trapezium rules that accounts for poles near the real axis, achieving exponential convergence and uniform high accuracy.

Findings

Achieves accuracy of 10^{-15} with only 12 quadrature points.

Maintains small relative errors for small and large arguments.

Provides explicit error bounds demonstrating exponential convergence.

Abstract

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel ({\em J. Math. Phys.} {\bf 34} (1956), 298--307) and Hunter and Regan ({\em Math. Comp.} {\bf 26} (1972), 539--541). We construct approximations which we prove are exponentially convergent as a function of $N$, the number of quadrature points, obtaining explicit error bounds which show that accuracies of $10^{-15}$ uniformly on the real line are achieved with N=12, this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.