Local Convergence Balls for Nonlinear Problems with Multiplicity and Their Extension to Eighth-Order Convergence

Ramandeep Behl,Fabricio Cevallos,Eulalia Martínez,Ali S Alshomrani

doi:10.1155/2019/1427809

Abstract

The main contribution of this study is to present a new optimal eighth-order scheme for locating zeros with multiplicity m≥1. An extensive convergence analysis is presented with the main theorem in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Moreover, a local convergence study for the optimal fourth-order method defined by the first two steps of the new method is presented, allowing us to obtain the radius of the local convergence ball. Finally, numerical tests on some real-life problems, such as a Van der Waals equation of state, a conversion chemical engineering problem, and two standard academic test problems, are presented, which confirm the theoretical results established in this paper and the efficiency of this proposed iterative method. We observed from the numerical experiments that our proposed iterative methods have good values for convergence radii. Further, they not only have faster convergence towards the desired zero of the involved function but also have both smaller residual error and a smaller difference between two consecutive iterations than current existing techniques.

Highlights

The construction of higher-order optimal multipoint iterative methods for locating multiple zeros with multiplicity m ≥ 1 of the involved function f is one of the toughest, most challenging, and most important tasks in the field of numerical analysis
Since the radius of local convergence for higherorder methods decreases with order, it is necessary to study its behavior when we present a new iterative method
We introduce a new idea for establishing local convergence results of iterative methods for locating multiple zeros, under the assumption of a bounding condition for the (m + 1)th derivative of the function f(x)

Summary

Introduction

The construction of higher-order optimal multipoint iterative methods for locating multiple zeros with multiplicity m ≥ 1 of the involved function f (where f : D ⊂ R 󳨀→ R is analytic in the enclosed region enclosing the required zero) is one of the toughest, most challenging, and most important tasks in the field of numerical analysis. In this paper the authors have constructed a new generic family of optimal eighth-order modified Newton-type multiple-zero finders and have studied their dynamical behavior; this kind of study was presented in [18]. None of these schemes have been studied from their local convergence treatment in Banach spaces. Our objective is to introduce methods for multiple roots of high order of convergence and to carry out a study of the local convergence of these For this purpose, our aim is to extend for the case of multiple roots, the optimal method of eighth order for simple roots of Chun and Neta [19].

Local Convergence of an Optimal Fourth-Order Scheme

Development of an Optimal Eighth-Order Scheme

Numerical Experiments

Methods

Conclusion