Stable high-order iterative methods for solving nonlinear models

Abstract

[EN] There are several problems of pure and applied science which can be studied in the unified framework of the scalar and vectorial nonlinear equations. In this paper, we propose a sixth-order family of Jarratt type methods for solving nonlinear equations. Further, we extend this family to the multidimensional case preserving the order of convergence. Their theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. We use complex dynamics techniques in order to select, among the elements of this class of iterative methods, those more stable. This process is made by analyzing the conjugacy class, calculating the fixed and critical points and getting conclusions from parameter and dynamical planes. For the implementation of the proposed schemes for system of nonlinear equations, we consider some applied science problems namely, Van der Pol problem, kinematic syntheses, etc. Further, we compare them with existing sixth-order methods to check the validity of the theoretical results. From the numerical experiments, we find that our proposed schemes perform better than the existing ones. Further, we also consider a variety of nonlinear equations to check the performance of the proposed methods for scalar equations.