An Extension of the Euler-Maclaurin Summation Formula to Nearly Singular Functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912526259716096 |
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| author | Wu, Bowei |
| author_facet | Wu, Bowei |
| contents | A extension of the Euler-Maclaurin (E-M) formula to near-singular functions is presented. This extension is derived based on earlier generalized E-M formulas for singular functions. The new E-M formulas consists of two components: a ``singular'' component that is a continuous extension of the earlier singular E-M formulas, and a ``jump'' component associated with the discontinuity of the integral with respect to a parameter that controls near singularity. The singular component of the new E-M formulas is an asymptotic series whose coefficients depend on the Hurwitz zeta function or the digamma function. Numerical examples of near-singular quadrature based on the extended E-M formula are presented, where accuracies of machine precision are achieved insensitive to the strength of the near singularity and with a very small number of quadrature nodes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Extension of the Euler-Maclaurin Summation Formula to Nearly Singular Functions Wu, Bowei Numerical Analysis 65D30, 65D32, 45E99 A extension of the Euler-Maclaurin (E-M) formula to near-singular functions is presented. This extension is derived based on earlier generalized E-M formulas for singular functions. The new E-M formulas consists of two components: a ``singular'' component that is a continuous extension of the earlier singular E-M formulas, and a ``jump'' component associated with the discontinuity of the integral with respect to a parameter that controls near singularity. The singular component of the new E-M formulas is an asymptotic series whose coefficients depend on the Hurwitz zeta function or the digamma function. Numerical examples of near-singular quadrature based on the extended E-M formula are presented, where accuracies of machine precision are achieved insensitive to the strength of the near singularity and with a very small number of quadrature nodes. |
| title | An Extension of the Euler-Maclaurin Summation Formula to Nearly Singular Functions |
| topic | Numerical Analysis 65D30, 65D32, 45E99 |
| url | https://arxiv.org/abs/2409.19192 |