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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10543_2025_1065_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$U'(a,z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10543_2025_1065_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$U'(a,z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in that region. Several tests show the accuracy and efficiency of the numerical algorithm.

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

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2025年4月18日

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

摘要

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 $U'(a,z)$ $U^{'} (a, z)$ 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 $U'(a,z)$ $U^{'} (a, z)$ in that region. Several tests show the accuracy and efficiency of the numerical algorithm.