On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866929665299447808 |
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| author | Canela, Jordi Evdoridou, Vasiliki Garijo, Antonio Jarque, Xavier |
| author_facet | Canela, Jordi Evdoridou, Vasiliki Garijo, Antonio Jarque, Xavier |
| contents | In this paper we study the dynamics of damped Traub's methods $T_δ$ when applied to polynomials. The family of damped Traub's methods consists of root finding algorithms which contain both Newton's ($δ=0$) and Traub's method ($δ=1$). Our goal is to obtain several topological properties of the basins of attraction of the roots of a polynomial $p$ under $T_1$, which are used to determine a (universal) set of initial conditions for which convergence to all roots of $p$ can be guaranteed. We also numerically explore the global properties of the dynamical plane for $T_δ$ to better understand the connection between Newton's method and Traub's method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub Canela, Jordi Evdoridou, Vasiliki Garijo, Antonio Jarque, Xavier Numerical Analysis Dynamical Systems 30D05, 37F10, 37F46 In this paper we study the dynamics of damped Traub's methods $T_δ$ when applied to polynomials. The family of damped Traub's methods consists of root finding algorithms which contain both Newton's ($δ=0$) and Traub's method ($δ=1$). Our goal is to obtain several topological properties of the basins of attraction of the roots of a polynomial $p$ under $T_1$, which are used to determine a (universal) set of initial conditions for which convergence to all roots of $p$ can be guaranteed. We also numerically explore the global properties of the dynamical plane for $T_δ$ to better understand the connection between Newton's method and Traub's method. |
| title | On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub |
| topic | Numerical Analysis Dynamical Systems 30D05, 37F10, 37F46 |
| url | https://arxiv.org/abs/2501.04450 |