Research Article An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics Rajni Sharma 1 and Ashu Bahl 2,3 1 Department of Applied Sciences, DAV Institute of Engineering and Technology, Kabir Nagar, Jalandhar 144008, India 2 Department of Mathematics, DAV College, Jalandhar 144008, India 3 I.K. Gujral Punjab Technical University, Kapurthala 144601, India Correspondence should be addressed to Ashu Bahl; bahl.ashu@redifmail.com Received 12 July 2015; Revised 7 October 2015; Accepted 12 October 2015 Academic Editor: Ying Hu Copyright © 2015 R. Sharma and A. Bahl. Tis 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. We present a new fourth order method for fnding simple roots of a nonlinear equation () = 0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its frst derivative. Terefore, the method has optimal order with efciency index 1.587 which is better than efciency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. Te conjugacy maps and extraneous fxed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. 1. Introduction Solving nonlinear equations is a common and important problem in science and engineering [1, 2]. Analytic methods for solving such equations are almost nonexistent and there- fore it is only possible to obtain approximate solutions by relying on numerical methods based on iterative procedures. With the advancement of computers, the problem of solving nonlinear equations by numerical methods has gained more importance than before. In this paper, we consider the problem of fnding simple root  of a nonlinear equation () = 0, where () is a continuously diferentiable function. Newton method is probably the most widely used algorithm for fnding simple roots, which starts with an initial approximation  0 closer to the root (say, ) and generates a sequence of successive iterates {  } ∞ 0 converging quadratically to simple roots (see [3]). It is given by  +1 =  − (  )   (  ) , =0,1,2,3, .... (1) In order to improve the order of convergence of Newton method, many higher order multistep methods [4] have been proposed and analyzed by various researchers at the expense of additional evaluations of functions, derivatives, and changes in the points of iterations. An extensive survey of the literature dealing with these methods of improved order is found in [3, 5, 6] and references therein. Euler method and Chebyshev method (see Traub [3]) Weerakoon and Fernando [7], Ostrowski’s square root method [5], Halley [8], Hansen and Patrick [9], and so forth are well-known third order methods requiring the evaluation of ,   , and   per step. Te famous Ostrowski’s method [5] is an important example of fourth order multipoint methods without memory. Te method requires two  and one   evaluations per step and is seen to be efcient compared to classical Newton method. Another well-known example of fourth order multipoint methods with same number of evaluations is King’s family of methods [10], which contains Ostrowski’s method as a special case. Chun et al. [11–13], Cordero et al. [14], and Kou et al. [15, 16] have also proposed fourth order methods requiring two  evaluations and one   evaluation per iteration. Jarratt [17] Hindawi Publishing Corporation Journal of Complex Analysis Volume 2015, Article ID 259167, 9 pages http://dx.doi.org/10.1155/2015/259167