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

Canela, Jordi; Evdoridou, Vasiliki; Garijo, Antonio and Jarque, Xavier (2023). On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub. Mathematische Zeitschrift, 303(3), article no. 55.

DOI: https://doi.org/10.1007/s00209-023-03215-8

Abstract

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 T₁, 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.

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