Local Convergence of an Efﬁcient High Convergence Order Method Using Hypothesis Only on the First Derivative

Ioannis Argyros,S.S Motsa,Ramandeep Behl

doi:10.3390/a8041076

Ioannis Argyros, S.S Motsa + Show 1 more

Open Access

https://doi.org/10.3390/a8041076

Copy DOI

Journal: Algorithms	Publication Date: Nov 20, 2015
Citations: 12	License type: CC BY 4.0

Affiliation: Cameron University, University of KwaZulu-Natal

Abstract

We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypotheses up to the fourth order derivative oreven higher of the function involved which restrict the applicability of the proposed scheme. However, only ﬁrst order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the ﬁrst order derivative. In this way,we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examplesare proposed to obtain the solutions of nonlinear equations. Our study does not exhibit thistype 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 guess x∗ that tells us how close the initial guess would be required for granted convergence of the Scheme (2)
Researchers mentioned that the initial guess should be close to the required root for the granted convergence of their proposed schemes for solving nonlinear equations

Summary

Introduction

Numerical analysis is a wide-ranging discipline having close connections with mathematics, computer science, engineering and the applied sciences. The results in [1], cannot apply to show the convergence of Scheme (2) or its special cases requiring hypotheses on the fifth derivative of function F or higher. In particular, there is a plethora of iterative methods for approximating solutions of nonlinear equations [1,2,3,4,5,6,7,8,10,11,12,13,14,15,16] These results show that initial guess should be close to the required root for the convergence of the corresponding methods. In the present study we expand the applicability of Scheme (2) using only hypotheses on the first order derivative of function F.

Local Convergence

Numerical Example and Applications

Conclusions