Journal of Applied Mathematics
Volume 2012 (2012), Article ID 497023, 24 pages
Research Article

Construction of Optimal Derivative-Free Techniques without Memory

1Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
2Allied Network for Policy Research and Advocacy for Sustainability, IEEE, Mauritius, Mauritius
3Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa
4School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg, South Africa

Received 5 July 2012; Accepted 3 September 2012

Academic Editor: Alicia Cordero

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.


Construction of iterative processes without memory, which are both optimal according to the Kung-Traub hypothesis and derivative-free, is considered in this paper. For this reason, techniques with four and five function evaluations per iteration, which reach to the optimal orders eight and sixteen, respectively, are discussed theoretically. These schemes can be viewed as the generalizations of the recent optimal derivative-free family of Zheng et al. in (2011). This procedure also provides an n-step family using function evaluations per full cycle to possess the optimal order 2n. The analytical proofs of the main contributions are given and numerical examples are included to confirm the outstanding convergence speed of the presented iterative methods using only few function evaluations. The second aim of this work will be furnished when a hybrid algorithm for capturing all the zeros in an interval has been proposed. The novel algorithm could deal with nonlinear functions having finitely many zeros in an interval.