Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914633386819584 |
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| author | Fornaess, John Erik Hu, Mi Truong, Tuyen Trung Watanabe, Takayuki |
| author_facet | Fornaess, John Erik Hu, Mi Truong, Tuyen Trung Watanabe, Takayuki |
| contents | A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author.
Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions, with BNQN. In general, they look more smooth than that of Newton's method.
In this paper, we continue to experimentally explore in depth this remarkable phenomenon, and connect BNQN to Newton's flow and Voronoi's diagram. This link poses a couple of challenging puzzles to be explained. Experiments also indicate that BNQN is more robust against random perturbations than Newton's method and Random Relaxed Newton's method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding Fornaess, John Erik Hu, Mi Truong, Tuyen Trung Watanabe, Takayuki Optimization and Control Machine Learning Numerical Analysis Complex Variables Dynamical Systems A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions, with BNQN. In general, they look more smooth than that of Newton's method. In this paper, we continue to experimentally explore in depth this remarkable phenomenon, and connect BNQN to Newton's flow and Voronoi's diagram. This link poses a couple of challenging puzzles to be explained. Experiments also indicate that BNQN is more robust against random perturbations than Newton's method and Random Relaxed Newton's method. |
| title | Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding |
| topic | Optimization and Control Machine Learning Numerical Analysis Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2401.01393 |