Some second-derivative-free variants of super-Halley method with fourth-order convergence 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 present some new variants of super-Halley method for solving nonlinear equations. These methods are free from second derivatives and require one function and two first derivative evaluations per iteration. Analysis of con- vergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Newton’s method; Iterative methods; Nonlinear equations; Order of convergence; Super-Halley method 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 [1]. A family of methods which accelerate Newton’s method, called super-Halley method [2,3] is given by x nþ1 ¼ x n  1 þ 1 2 L f ðx n Þ 1  L f ðx n Þ   f ðx n Þ f 0 ðx n Þ ; ð2Þ where L f ðx n Þ¼ f ðx n Þf 00 ðx n Þ f 0 ðxn Þ 2 . This family is known to be third-order, but it depends on second derivative, so that its use is severely restricted in applications from a practical point of view. Therefore, it is important and interesting to develop 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.003 * 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 195 (2008) 537–541 www.elsevier.com/locate/amc