Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

Alicia Cordero,Juan R Torregrosa,Javier G Maimó,María P Vassileva

doi:10.3390/axioms8020051

Abstract

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.

Highlights

Systems of nonlinear equations must usually be solved when nonlinear models, appearing in Science and Engineering, are discretized
There are no analytical techniques for solving these systems, so we approach their solutions by using iterative schemes
The most known iterative procedure is Newton’s scheme, in recent years, the focus of this area of research has been in constructing new iterative methods, trying to improve Newton’s one, in terms of convergence, efficiency, and stability

Summary

Introduction

Systems of nonlinear equations must usually be solved when nonlinear models, appearing in Science and Engineering, are discretized. In 2016, the authors of [14] proposed a parametric class of iterative schemes with fourth-order of convergence, including a very efficient fifth-order procedure. The efficiency index used is defined as p1/d+op , where p is the order of convergence of the method, d the number of functional evaluations per iteration, and op the number of products/quotients used for obtaining a new iterate. This class combined three evaluations of the nonlinear function and only one of the Jacobian matrix, per iteration

Objectives

Methods

Conclusion