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.

Figures (6)

• Figure 1: Test of the accuracy obtained with the Liouville-Green expansions. Left: Points where the error when testing the recurrence relation NIST:DLMF for $U(20,z)$ is smaller than $5\times 10^{-13}$. Right: Points where the error when testing the recurrence relations NIST:DLMF and NIST:DLMF for $U'(20,z)$ is smaller than $5\times 10^{-13}$.
• Figure 2: Left: Zeros obtained with the function zerosUaz(a,L) for $a=-3.2$ and $L=5$. Right: Estimated relative errors obtained.
• Figure 3: Left: Zeros obtained with the function zerosUaz(a,L) for $a=-13.1$ and $L=15$. Right: Estimated relative errors obtained.
• Figure 4: Left: Zeros obtained with the function zerosUaz(a,L) for $a=1.3$ and $L=10$. Right: Estimated relative errors obtained.
• Figure 5: Left: Zeros obtained with the function zerosUaz(a,L) for $a=10.7$ and $L=15$. Right: Estimated relative errors obtained.