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

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.