A family of composite 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 a family of new fourth-order iterative methods for solving nonlinear equations. Per iteration the methods consisting of the family require only two evaluations of the function and one evaluation of its derivative. Sev- eral numerical examples are given to illustrate the efficiency and performance of some of the presented methods. Ó 2006 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(a) = 0 and f 0 (a) 5 0, of a non- linear equation f(x) = 0. 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. There exist many iterative methods improving Newton’s method for solving nonlinear equations. However, many of those iterative methods depend on the second or higher derivatives in computing process which make their practical application restricted strictly. As a result, Newton’s method is frequently and alternatively used to solve nonlinear equations because of higher computational efficiency. In recent years, there has been some progress on iterative methods improving Newton’s method with cubic convergence that do not require the computation of second derivatives for solving nonlinear equations, see [1–9] and the reference therein. Furthermore, in [10–12] several iterative fourth-order methods which are free from second derivatives and that do require only three evaluations of both the function and its derivatives are proposed. 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.009 E-mail address: cbchun@kut.ac.kr Applied Mathematics and Computation 187 (2007) 951–956 www.elsevier.com/locate/amc