Some variants of King’s fourth-order family of methods for nonlinear equations Changbum Chun School of Liberal Arts, Korea University of Technology and Education, Cheonan, Chungnam 330-708, Republic of Korea Abstract In this paper, we present fourth-order families of new variants of King’s fourth-order family of methods for solving nonlinear equations. Per iteration the obtained methods require two evaluations of the given function and one of its first derivative. The classical Traub–Ostrowski method is obtained as a special variant of King’s method. Several numerical examples are given to illustrate the efficiency and performance of the presented methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Newton’s method; King’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. 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 methods have been proposed and analyzed for solving nonlin- ear equations that improve several classical methods such as Newton’s method, Halley-like methods etc., and do require three evaluations of the given function and its first derivative per iteration, see [2–10] and the ref- erence therein. It has been shown that these methods are efficient in their performance, can compete with New- ton’s method as well as the classical methods such as Jarratt’s method [11], Traub–Ostrowski’ method [12]. In this paper, we also present and analyze new fourth-order families of methods for solving nonlinear equa- tions, which are obtained as variants of King’s fourth-order family [6] defined by w n ¼ x n  f ðx n Þ f 0 ðx n Þ ; ð2Þ 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.006 E-mail address: cbchun@kut.ac.kr Applied Mathematics and Computation 190 (2007) 57–62 www.elsevier.com/locate/amc