Local Convergence of an Optimal Eighth Order Method under Weak Conditions

Ioannis Argyros,Ramandeep Behl,S.S Motsa

doi:10.3390/a8030645

Abstract

We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence under hypotheses on the seventh derivative or even higher, although only the first derivative and the divided difference appear in these methods. The convergence in this study is shown under hypotheses only on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

Highlights

We are concerned with the problem of approximating a locally-unique solution x∗
The semi-local convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of iteration procedures
We study the local convergence of the three-step method defined for each n = 0, 1, 2, . . . by: yn

Summary

Introduction

We are concerned with the problem of approximating a locally-unique solution x∗. We provide the radius of the convergence ball, computable error bounds on the distances involved and the uniqueness of the solution result using Lipschitz constants Such results were not given in [1] or the earlier related studies [8,9,10,11,12].

Local Convergence

Numerical Example and Applications