Generalizing Traub's method to a parametric iterative class for solving multidimensional nonlinear problems

Abstract

[EN] In this work, we modify the iterative structure of Traub's method to include a real parameter alpha$$ \alpha $$. A parametric family of iterative methods is obtained as a generalization of Traub, which is also a member of it. The cubic order of convergence is proved for any value of alpha$$ \alpha $$. Then, a dynamical analysis is performed after applying the family for solving a system cubic polynomials by means of multidimensional real dynamics. This analysis allows to select the best members of the family in terms of stability as a preliminary study to be generalized to any nonlinear function. Finally, some iterative schemes of the family are used to check numerically the previous developments when they are used to approximate the solutions of academic nonlinear problems and a chemical diffusion reaction problem.