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

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 $\mathrm{\mid }a\mathrm{\mid }$ small, the asymptotic approximations are complemented with a few fixed point iterations requiring the evaluation of $U(a,z)$ and $U^{\mathrm{\prime }}(a,z)$ 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 $U^{\mathrm{\prime }}(a,z)$ in that region. Several tests show the accuracy and efficiency of the numerical algorithm.