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The Scientific World Journal
Volume 2014 (2014), Article ID 134673, 9 pages
http://dx.doi.org/10.1155/2014/134673
Research Article

On a New Three-Step Class of Methods and Its Acceleration for Nonlinear Equations

1Department of Mathematics, Islamic Azad University, Hamedan Branch, Hamedan, Iran
2Department of Mathematics, Islamic Azad University, Shahrekord Branch, Shahrekord, Iran
3Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, Thohoyandou 0950, South Africa

Received 20 July 2014; Accepted 16 August 2014; Published 3 September 2014

Academic Editor: Hassan Saberi Nik

Copyright © 2014 T. Lotfi et al. 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.

Abstract

A class of derivative-free methods without memory for approximating a simple zero of a nonlinear equation is presented. The proposed class uses four function evaluations per iteration with convergence order eight. Therefore, it is an optimal three-step scheme without memory based on Kung-Traub conjecture. Moreover, the proposed class has an accelerator parameter with the property that it can increase the convergence rate from eight to twelve without any new functional evaluations. Thus, we construct a with memory method that increases considerably efficiency index from to . Illustrations are also included to support the underlying theory.