Dynamics and Stability on a Family of Optimal Fourth-Order Iterative Methods

Abstract

In this manuscript, we propose a parametric family of iterative methods of fourth-order convergence, and the stability of the class is studied through the use of tools of complex dynamics. We obtain the fixed and critical points of the rational operator associated with the family. A stability analysis of the fixed points allows us to find sets of values of the parameter for which the behavior of the corresponding method is stable or unstable; therefore, we can select the regions of the parameter in which the methods behave more efficiently when they are applied for solving nonlinear equations or the regions in which the schemes have chaotic behavior.