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arXiv · 2501.04450

On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub

Jordi Canela·Vasiliki Evdoridou·Antonio Garijo·Xavier Jarque
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Abstract

In this paper we study the dynamics of damped Traub's methods $T_\delta$ when applied to polynomials. The family of damped Traub's methods consists of root finding algorithms which contain both Newton's ($\delta=0$) and Traub's method ($\delta=1$). 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 $T_\delta$ to better understand the connection between Newton's method and Traub's method.

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Jordi Canela, Vasiliki Evdoridou, Antonio Garijo, Xavier Jarque. 2025-01-08. On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub. https://doi.org/10.1007/s00209-023-03215-8

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