2024/12/17 by T. M. Dunster, Dunster, T. M., Amparo Gil +5
Computer Science · Engineering · Mathematics · #33B15 #33C15 #65D20 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Classical Analysis and ODEs (math.CA) #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2412.13085
openalex publication_date 2024/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.