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Discrete Dynamics in Nature and Society
Volume 2013, Article ID 301718, 10 pages
Research Article

Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

1Department of Chemistry, Roudehen Branch, Islamic Azad University, Roudehen, Iran
2Department of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan, Iran
3Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa

Received 4 January 2013; Accepted 2 April 2013

Academic Editor: Fathi Allan

Copyright © 2013 Farahnaz Soleimani 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.


First, we make the Jain's derivative-free method optimal and subsequently increase its efficiency index from 1.442 to 1.587. Then, a novel three-step computational family of iterative schemes for solving single variable nonlinear equations is given. The schemes are free from derivative calculation per full iteration. The optimal family is constructed by applying the weight function approach alongside an approximation for the first derivative of the function in the last step in which the first two steps are the optimized derivative-free form of Jain's method. The convergence rate of the proposed optimal method and the optimal family is studied. The efficiency index for each method of the family is 1.682. The superiority of the proposed contributions is illustrated by solving numerical examples and comparing them with some of the existing methods in the literature. In the end, we provide the basins of attraction for some methods to observe the beauty of iterative nonlinear solvers in providing fractals and also choose the best method in case of larger attraction basins.