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.

翻译：本文提出了一种数值算法（在Matlab中实现），用于计算复平面上抛物柱函数$U(a,z)$在特定区域内的零点。该算法首先采用高精度近似计算第一个零点，随后基于四阶不动点方法，结合泰勒级数计算抛物柱函数并精心选择步长，高效地计算其余零点。对于$|a|$较小的情况，在复零点所在区域通过少量不动点迭代（需计算$U(a,z)$和$U'(a,z)$的值）对渐近近似进行补充。为提升该区域$U(a,z)$和$U'(a,z)$的计算效率，本文推导了Liouville-Green展开式。多项测试验证了该数值算法的精确性与高效性。