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Open AccessArticle

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

Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa
Author to whom correspondence should be addressed.
Academic Editor: Alicia Cordero
Algorithms 2015, 8(4), 1076-1087;
Received: 25 September 2015 / Accepted: 11 November 2015 / Published: 20 November 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
PDF [234 KB, uploaded 20 November 2015]


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. View Full-Text
Keywords: Newton-like method; local convergence; efficiency index; optimum method Newton-like method; local convergence; efficiency index; optimum method
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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MDPI and ACS Style

Argyros, I.K.; Behl, R.; Motsa, S. Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative. Algorithms 2015, 8, 1076-1087.

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