Vectorial Iterative Schemes with Memory for Solving Nonlinear Systems of Equations ^†

Abstract

There exist in the literature many iterative methods for solving nonlinear problems. Some of these methods can be transferred directly to the context of nonlinear systems, keeping the order of convergence, but others cannot be directly extended to a multidimensional case. Sometimes, the procedures are designed specifically for multidimensional problems by using different techniques, as composition and reduction or weight-function procedures, among others. Our main aim is not only to design an iterative scheme for solving nonlinear systems but also to assure its high order of convergence by means of the introduction of matrix accelerating parameters. This is a challenging area of numerical analysis wherein there are still few procedures defined. Once the iterative method has been designed, it is necessary to carry out a dynamical study in order to verify the wideness of the basins of attraction of the roots and compare its stability with other known methods.