Ball comparison between frozen Potra and Schmidt–Schwetlick schemes with dynamical analysis

Abstract

In this article, we propose a new research related to the convergence of the frozen Potra and Schmidt–Schwetlick schemes when we apply to equations. The purpose of this study is to introduce a comparison between two solutions to equations under the same conditions. In particular, we show the convergence radius and the uniqueness ball coincidence, while the error estimates are generally different. In this work, we extended the local convergence for Banach space valued operators using only the divided difference of order one and the first derivative of the schemes. This is a great advantage since we improve convergence by avoiding calculating higher-order derivatives that can either be difficult or not even exist. On the other hand, we also present a dynamical study of the behavior of a method compared with its no frozen alternative in order to see the behavior of both. We will study the basins of attraction of the two methods to three different polynomials involving two real, three real, and two real and two complex different solutions.