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

Abstract

AbstractIn this paper we study the dynamics of damped Traub’s methods$$T_\delta $$Tδwhen applied to polynomials. The family of damped Traub’s methods consists of root finding algorithms which contain both Newton’s ($$\delta =0$$δ=0) and Traub’s method ($$\delta =1$$δ=1). Our goal is to obtain several topological properties of the basins of attraction of the roots of a polynomialpunder$$T_1$$T1, which are used to determine a (universal) set of initial conditions for which convergence to all roots ofpcan be guaranteed. We also numerically explore the global properties of the dynamical plane for$$T_\delta $$Tδto better understand the connection between Newton’s method and Traub’s method.