Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations
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arXiv
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| Autori principali: | , , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
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| _version_ | 1866909363718848512 |
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| author | Parkinson, Erik Wall, Kate Slagle, Jane Treuhaft, Daniel de la Bruere, Xander Goldrup, Samuel Keith, Timothy Call, Peter Jarvis, Tyler J. |
| author_facet | Parkinson, Erik Wall, Kate Slagle, Jane Treuhaft, Daniel de la Bruere, Xander Goldrup, Samuel Keith, Timothy Call, Peter Jarvis, Tyler J. |
| contents | We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available on GitHub. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations Parkinson, Erik Wall, Kate Slagle, Jane Treuhaft, Daniel de la Bruere, Xander Goldrup, Samuel Keith, Timothy Call, Peter Jarvis, Tyler J. Numerical Analysis Algebraic Geometry 65H10 We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available on GitHub. |
| title | Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations |
| topic | Numerical Analysis Algebraic Geometry 65H10 |
| url | https://arxiv.org/abs/2401.02114 |