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Open AccessArticle

Local Convergence of an Optimal Eighth Order Method under Weak Conditions

Cameron University, Department of Mathematics Sciences Lawton, OK 73505, USA
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa
Author to whom correspondence should be addressed.
Academic Editor: Alicia Cordero
Algorithms 2015, 8(3), 645-655;
Received: 9 June 2015 / Revised: 31 July 2015 / Accepted: 5 August 2015 / Published: 19 August 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
PDF [236 KB, uploaded 19 August 2015]


We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence under hypotheses on the seventh derivative or even higher, although only the first derivative and the divided difference appear in these methods. The convergence in this study is shown under hypotheses only on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply. 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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Argyros, I.K.; Behl, R.; Motsa, S. Local Convergence of an Optimal Eighth Order Method under Weak Conditions. Algorithms 2015, 8, 645-655.

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