A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913758836686848 |
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| author | Dunster, T. M. Gil, A. Ruiz-Antolín, D. Segura, J. |
| author_facet | Dunster, T. M. Gil, A. Ruiz-Antolín, D. Segura, J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane Dunster, T. M. Gil, A. Ruiz-Antolín, D. Segura, J. Numerical Analysis Classical Analysis and ODEs 33B15, 33C15, 65D20 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. |
| title | A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane |
| topic | Numerical Analysis Classical Analysis and ODEs 33B15, 33C15, 65D20 |
| url | https://arxiv.org/abs/2412.13085 |