Some fourth-order modifications of Newton’s method Changbum Chun a, * , YoonMee Ham b a School of Liberal Arts, Korea University of Technology and Education, Cheonan, Chungnam 330-708, Republic of Korea b Department of Mathematics Kyonggi University, Suwon 443-760, Republic of Korea Abstract In this paper, we construct some fourth-order modifications of Newton’s method for solving nonlinear equations. Any two existing fourth-order methods can be effectively used to give rise to new fourth-order methods. Per iteration the new methods require two evaluations of the function and one of its first-derivative. Numerical examples are given 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; Root-finding methods 1. Introduction One of the most important problems in numerical analysis is that of solving nonlinear equations. To solve nonlinear equations, iterative methods such as Newton’s method are usually used. Throughout we consider iterative methods to find a simple root a, i.e., f(a) = 0 and f 0 (a) 5 0, of a nonlinear equation f(x) = 0 that uses f and f 0 but not the higher derivatives of f. Newton’s method for the calculation of a is probably the most widely used iterative methods defined by x nþ1 ¼ x n  f ðx n Þ f 0 ðx n Þ : ð1Þ It is well known [1] that this method is quadratically convergent. In recent years, many efficient at least fourth-order modifications of the Newton method that do not require the computation of second derivatives have been developed and analyzed in open literature, see [2–10] and references therein. Some of those methods, however, often depend on the first derivative at another point iter- ated by Newton’s method or the other classical method. So, it would be desirable to remove that dependence from such methods while either keeping the order of convergence or increasing the efficiency index, so their practical utility increase, this is the main purpose of this work. 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.08.003 * Corresponding author. E-mail addresses: cbchun@kut.ac.kr (C. Chun), ymham@kyonggi.ac.kr (Y. Ham). Available online at www.sciencedirect.com Applied Mathematics and Computation 197 (2008) 654–658 www.elsevier.com/locate/amc