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

Algorithms

Ioannis K. Argyros¹  Ramandeep Behl²  S.S. Motsa² 

[1] Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA; E-Mail:;School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa; E-Mail:

关键词: Newton-like method;    local convergence;    efficiency index;    optimum method;   

DOI  :  10.3390/a8041076

来源: mdpi

【 摘 要 】

We present a local convergence analysis of an eighth order three step method in order to approximate a locally unique solution of nonlinear equation in a Banach space setting. In an earlier study by Sharma and Arora (2015), 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 first order derivative appears in the proposed scheme. In order to overcome this 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 examples are proposed to obtain the solutions of nonlinear equations. Our study does not exhibit this type of problem/restriction.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.