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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 318165, 14 pages
http://dx.doi.org/10.1155/2012/318165
Research Article

Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods

1Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
2Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa

Received 21 August 2012; Accepted 20 September 2012

Academic Editor: Turgut Öziş

Copyright © 2012 F. Soleymani and S. Shateyi. 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

Optimization problems defined by (objective) functions for which derivatives are unavailable or available at an expensive cost are emerging in computational science. Due to this, the main aim of this paper is to attain as high as possible of local convergence order by using fixed number of (functional) evaluations to find efficient solvers for one-variable nonlinear equations, while the procedure to achieve this goal is totally free from derivative. To this end, we consider the fourth-order uniparametric family of Kung and Traub to suggest and demonstrate two classes of three-step derivative-free methods using only four pieces of information per full iteration to reach the optimal order eight and the optimal efficiency index 1.682. Moreover, a large number of numerical tests are considered to confirm the applicability and efficiency of the produced methods from the new classes.