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arXiv · 2412.13085

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

T. M. Dunster·A. Gil·D. Ruiz-Antolín·J. Segura
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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.

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T. M. Dunster, A. Gil, D. Ruiz-Antolín, J. Segura. 2024-12-17. A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane. https://arxiv.org/abs/2412.13085

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