Some fourth-order iterative methods for solving nonlinear equations Changbum Chun School of Liberal Arts, Korea University of Technology and Education, Cheonan City, Chungnam 330-708, Republic of Korea Abstract In this paper we present some fourth-order iterative methods for solving nonlinear equations, which contains the well- known King’s fourth-order family as a particular case. Per iteration the methods require two function and one first deriv- ative evaluations. Numerical comparisons are made with several other existing methods to show the performance of the presented methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Newton’s method; Iterative methods; Nonlinear equations; Order of convergence 1. Introduction In this paper, we consider iterative methods to find a simple root a, i.e., f ðaÞ¼ 0 and f 0 ðaÞ 6¼ 0, of a non- linear equation f ðxÞ¼ 0. The case for multiple roots will not be considered in the present contribution. Newton’s method is the well-known iterative method for finding a by using x nþ1 ¼ x n  f ðx n Þ f 0 ðx n Þ ð1Þ that converges quadratically in some neighborhood of a [1]. In recent years, some fourth-order iterative meth- ods have been proposed and analyzed for solving nonlinear equations which improve such classical methods as Newton’s method, Halley-like methods, etc. in a number of ways. They usually require three evaluations of the given function and its first derivative per iteration, see [2–9] and the references therein. These methods can also be viewed as obtained by taking an appropriate approximation to f 0 ðw n Þ in the following iteration scheme: x nþ1 ¼ w n  f ðw n Þ f 0 ðw n Þ ; ð2Þ where w n ¼ /ðx n Þ, /ðxÞ is usually an iteration function such as the Newton iteration function. 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.105 E-mail address: cbchun@kut.ac.kr Available online at www.sciencedirect.com Applied Mathematics and Computation 195 (2008) 454–459 www.elsevier.com/locate/amc