Multidimensional stability analysis of a family of bi-parametric iterative methods

Abstract

[EN] In this paper, we present a multidimensional real dynamical study of the Ostrowsky-Chun family of iterative methods to solve systems of nonlinear equations. This family was defined initially for solving scalar equations but, in general, scalar methods can be transferred to make them suitable to solve nonlinear systems. The complex dynamical behavior of the rational operator associated to a scalar method applied to low-degree polynomials has shown to be an efficient tool for analyzing the stability and reliability of the methods. However, a good scalar dynamical behavior does not guarantee a good one in multidimensional case. We found different real intervals where both parameters can be defined assuring a completely stable performance and also other regions where it is dangerous to select any of the parameters, as undesirable behavior as attracting elements that are not solution of the problem to be solved appear. This performance is checked on a problem of chemical wave propagation, Fisher's equation, where the difference in numerical results provided by those elements of the class with good stability properties and those showed to be unstable, is clear.