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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1903.10697 |
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| _version_ | 1866915498162126848 |
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| author | DeFranco, Mario |
| author_facet | DeFranco, Mario |
| contents | We present generalizations of the Newton-Raphson-Simpson method. Specifically, for a positive integer $m$ and the sequence of coefficients of a Taylor series of a function $f(z)$, we define an algorithm we denote by NRS($m$) which is a way to evaluate, in our terminology, a sum of $m$ formal zeros of $f(z)$. We prove that NRS(1) yields the familiar iterations of the Newton-Raphson-Simpson method. We also prove that NRS($m$) is way to evaluate certain $\mathscr{A}$-hypergeometric series defined by Sturmfels. In order to define these algorithms, we make use of combinatorial objects which we call trees with negative vertex degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1903_10697 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On Generalizations of the Newton-Raphson-Simpson Method DeFranco, Mario Combinatorics We present generalizations of the Newton-Raphson-Simpson method. Specifically, for a positive integer $m$ and the sequence of coefficients of a Taylor series of a function $f(z)$, we define an algorithm we denote by NRS($m$) which is a way to evaluate, in our terminology, a sum of $m$ formal zeros of $f(z)$. We prove that NRS(1) yields the familiar iterations of the Newton-Raphson-Simpson method. We also prove that NRS($m$) is way to evaluate certain $\mathscr{A}$-hypergeometric series defined by Sturmfels. In order to define these algorithms, we make use of combinatorial objects which we call trees with negative vertex degree. |
| title | On Generalizations of the Newton-Raphson-Simpson Method |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1903.10697 |