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Journal of Applied Mathematics
Volume 2012, Article ID 568740, 15 pages
http://dx.doi.org/10.1155/2012/568740
Research Article

A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order

1Young Researchers Club, Islamic Azad University, Zahedan Branch, Zahedan, Iran
2Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran

Received 8 October 2011; Accepted 1 November 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 F. Soleymani 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 three-step eighth-order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation per full iteration, which is important in engineering problems. One method of the class is established analytically. To test the derived methods from the class, we apply them to a lot of nonlinear scalar equations. Numerical examples suggest that the novel class of derivative-free methods is better than the existing methods of the same type in the literature.