A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane

TL;DR

This paper presents a Matlab algorithm for accurately computing the complex zeros of parabolic cylinder functions using asymptotic approximations, fixed point methods, and Liouville-Green expansions, demonstrating high efficiency and precision.

Contribution

The paper introduces a novel numerical algorithm combining asymptotic approximations and fixed point iterations for zeros of parabolic cylinder functions in the complex plane.

Findings

Algorithm achieves high accuracy in zero computation

Efficient for small and large parameter values

Validated through multiple numerical tests

Abstract

A numerical algorithm (implemented in Matlab) for computing the zeros of the parabolic cylinder function $U(a,z)$ in domains of the complex plane is presented. The algorithm uses accurate approximations to the first zero plus a highly efficient method based on a fourth-order fixed point method with the parabolic cylinder functions computed by Taylor series and carefully selected steps, to compute the rest of the zeros. For $|a|$ small, the asymptotic approximations are complemented with a few fixed point iterations requiring the evaluation of $U(a,z)$ and $U'(a,z)$ in the region where the complex zeros are located. Liouville-Green expansions are derived to enhance the performance of a computational scheme to evaluate $U(a,z)$ and $U'(a,z)$ in that region. Several tests show the accuracy and efficiency of the numerical algorithm.