Stability of King’s family of iterative methods with memory

Abstract

In the literature exist many iterative methods with memory for solving nonlinear equations, the most of them designed in the last years. As they use the information of (at least) the two previous iterates to generate the new one, usual techniques of complex dynamics are not useful in this case. In this paper, we present some real multidimensional dynamical tools to undertake this task, applied on a very well-known family of iterative schemes; King’s class. It is showed that the most of elements of this class present a very stable behavior, visualized in different dynamical planes. However, pathological cases as attracting strange fixed points or periodic orbits can also be found.