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

TL;DR

This paper investigates the dynamics and basins of attraction of a family of root-finding algorithms that interpolate between Newton's and Traub's methods, providing insights into their convergence properties.

Contribution

It characterizes the topological properties of basins of attraction for the damped Traub's methods and identifies initial conditions guaranteeing convergence to polynomial roots.

Findings

Topological analysis of basins of attraction for $T_1$

Universal initial conditions for convergence

Numerical exploration of dynamical planes

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.