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Journal of Applied Mathematics
Volume 2012, Article ID 958020, 13 pages
Research Article

Computing Simple Roots by an Optimal Sixteenth-Order Class

1Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
2Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa
3Department of Civil Engineering, Islamic Azad University, Zahedan Branch, Zahedan, Iran

Received 21 August 2012; Revised 6 October 2012; Accepted 7 October 2012

Academic Editor: Changbum Chun

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.


The problem considered in this paper is to approximate the simple zeros of the function by iterative processes. An optimal 16th order class is constructed. The class is built by considering any of the optimal three-step derivative-involved methods in the first three steps of a four-step cycle in which the first derivative of the function at the fourth step is estimated by a combination of already known values. Per iteration, each method of the class reaches the efficiency index , by carrying out four evaluations of the function and one evaluation of the first derivative. The error equation for one technique of the class is furnished analytically. Some methods of the class are tested by challenging the existing high-order methods. The interval Newton's method is given as a tool for extracting enough accurate initial approximations to start such high-order methods. The obtained numerical results show that the derived methods are accurate and efficient.