Stability Analysis of Iterative Methods for Solving Nonlinear Algebraic Systems

In this chapter, we present a multidimensional real dynamical analysis of a new class of iterative method for approximating the solutions of nonlinear systems of algebraic equations. With the use of the well known discrete dynamic multivariate tools, we study the behavior of the multidimensional rational operator associated with the iterative method, acting on a system of quadratic polynomials of separate and mixed variables, respectively. Some results about the stability of the proposed class are presented. These results allow us to detect and avoid the elements of the family with bad stability properties and chaotical behaviour. Some numerical tests are presented for confirming the theoretical and dynamical results.