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Algorithms 2016, 9(4), 65; doi:10.3390/a9040065

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

1
Department of Mathematics Sciences Lawton, Cameron University, Lawton, OK 73505, USA
2
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg 3209, South Africa
*
Author to whom correspondence should be addressed.
Academic Editor: Alicia Cordero
Received: 6 May 2016 / Revised: 20 September 2016 / Accepted: 20 September 2016 / Published: 29 September 2016
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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. View Full-Text
Keywords: Kung-Traub method; local convergence; divided difference; Banach space; Lipschitz constant; radius of convergence Kung-Traub method; local convergence; divided difference; Banach space; Lipschitz constant; radius of convergence
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Argyros, I.K.; Behl, R.; Motsa, S.S. Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative. Algorithms 2016, 9, 65.

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