Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 425867, 9 pages doi:10.1155/2012/425867 Research Article A Family of Three-Point Methods of Ostrowski’s Type for Solving Nonlinear Equations Jovana D ˇ zuni ´ c and Miodrag S. Petkovi´ c Department of Mathematics, Faculty of Electronic Engineering, University of Niˇ s, 18000 Niˇ s, Serbia Correspondence should be addressed to Jovana Dˇ zuni´ c, jovana.dzunic@elfak.ni.ac.rs Received 6 October 2011; Accepted 24 November 2011 Academic Editor: Vu Phat Copyright q 2012 J. D ˇ zuni´ c and M. S. Petkovi´ c. 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. A class of three-point methods for solving nonlinear equations of eighth order is constructed. These methods are developed by combining two-point Ostrowski’s fourth-order methods and a modified Newton’s method in the third step, obtained by a suitable approximation of the first derivative using the product of three weight functions. The proposed three-step methods have order eight costing only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Two numerical examples for various weight functions are given to demonstrate very fast convergence and high computational efficiency of the proposed multipoint methods. 1. Introduction Multipoint methods for solving nonlinear equations f x 0, where f : D ⊂ R → R, possess an important advantage since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. More details may be found in the book 1 and many papers published in the first decade of the 21st century. In this paper we present a new family of three-point methods which employs Ostrowski’s method in the first two steps and suitably chosen weight functions in the third step. The order of this family is eight requiring four function evaluations. We start with a three-step scheme omitting iteration index for simplicity y  x - f x f ′ x , z  y - f ( y ) f ′ x · f x f x - 2f ( y ) ,  x  z - f z f ′ z , 1.1