New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis

Abstract

In this paper, a two-step class of fourth-order iterative methods for solving systems of nonlinear equations is presented. We further extend the two-step class to establish a new sixth-order family which requires only one additional functional evaluation. The convergence analysis of the proposed classes is provided under several mild conditions. A complete dynamical analysis is made, by using real multidimensional discrete dynamics, in order to select the most stable elements of both families of fourth- and sixth-order of convergence. To get this aim, a novel tool based on the existence of critical points has been used, the parameter line. The analytical discussion of the work is upheld by performing numerical experiments on some application-oriented problems. We provide an implementation of the proposed scheme on nonlinear optimization problem and zero-residual nonlinear least-squares problems taken from the constrained and unconstrained testing environment test set. Finally, based on numerical results, it has been concluded that our methods are comparable with the existing ones of similar nature in terms of order, efficiency, and computational time and also that the stability results provide the most efficient member of each class of iterative schemes.