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

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.