Dynamical Aspects of Some Convex Acceleration Methods as Purely Iterative Algorithm for Newton S Maps

In this paper we define purely iterative algorithm for Newton s maps which is a slight modification of the concept of purely iterative algorithm due to Smale. For this, we use a characterization of rational maps which arise from Newton s method applied to complex polynomials. We prove the Scaling Theorem for purely iterative algorithm for Newton s map. Then we focus our study in dynamical aspects of three root-finding iterative methods viewed as a purely iterative algorithm for Newton s map: Whittaker s iterative method, the super-Halley iterative method and a modification of the latter. We give a characterization of the attracting fixed points which correspond to the roots of a polynomial. Also, numerical examples are included in order to show how to use the characterization of fixed points. Finally, we give a description of the parameter spaces of the methods under study applied to a one-parameter family of generic cubic polynomials. © 2014 Elsevier Inc.