Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative

Ioannis Argyros,Ramandeep Behl,Sandile Motsa

doi:10.3390/a9040065

Ioannis Argyros, Ramandeep Behl + Show 1 more

Open Access

https://doi.org/10.3390/a9040065

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Journal: Algorithms	Publication Date: Sep 29, 2016
Citations: 1	License type: CC BY 4.0

Affiliation: Cameron University, University of KwaZulu-Natal

Abstract

In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not discuss these properties. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only the first order derivatives appear in the proposed scheme. To overcome this problem, we present the hypotheses for the proposed scheme maximum up to first order derivative. In this way, we not only expand the applicability of the methods but also suggest convergence domain. Finally, a variety of concrete numerical examples are proposed where earlier studies can not be applied to obtain the solutions of nonlinear equations on the other hand our study does not exhibit this type of problem/restriction.

Highlights

Numerical analysis is a wide-ranging discipline having close connections with mathematics, computer science, engineering and the applied sciences
We further present the range of initial guesses x ∗ that tell us how close the initial guesses should be required to guarantee the convergence of the method Equation (2)
We provide the radius of convergence for the considered test problem which gives the guarantee for the convergence

Summary

Introduction

Numerical analysis is a wide-ranging discipline having close connections with mathematics, computer science, engineering and the applied sciences. In particular there is a plethora of iterative methods for approximating solutions of nonlinear equations [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] These results show that initial guesses should be close to the required root for the convergence of the corresponding methods. We expand the applicability of method given by Equation (2) using only hypotheses on the first order derivative of function F. We further present the range of initial guesses x ∗ that tell us how close the initial guesses should be required to guarantee the convergence of the method Equation (2) This problem was not addressed in [13].

Local Convergence

Numerical Example and Applications

Methods n xn

Results and Discussion

Basin of Attractions

Conclusions