A new modified Halley method without second derivatives for nonlinear equation Muhammad Aslam Noor * , Waseem Asghar Khan, Akhtar Hussain Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract In a recent paper, Noor and Noor [K. Inayat Noor, M. Aslam Noor, Predictor–corrector Halley method for nonlinear equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.11.023] have suggested and analyzed a predictor–corrector method Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method is a robust one. Several examples are given to illus- trate the efficiency and the performance of this new method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Halley method; Predictor-corrector method; Iterative methods; Convergence; Examples 1. Introduction In recent years, several iterative type methods have been developed by using the Taylor series, decomposi- tion and quadrature formulae, see [1–11] and the references therein. Using the technique of updating the solu- tion and Taylor series expansion, Noor and Noor [10] have suggested and analyzed a sixth-order predictor– corrector iterative type Halley method for solving the nonlinear equations. Also Kou et al. [6,7] have also sug- gested a class of fifth-order iterative methods. In the implementation of these methods, one has to evaluate the second derivative of the function, which is a serious drawback of these methods. To overcome these draw- backs, we modify the predictor–corrector Halley method by replacing the second derivatives of the function f by its finite difference scheme. We prove that the new modified predictor–corrector method is of fifth-order convergence. We also present the comparison of the new method with the methods of Kou et al. [6,7]. In pass- ing, we would like to point out the results presented by Kou et al. [6,7] are incorrect. We also rectify this error. Several examples are given to illustrate the efficiency and robustness of the new proposed method. 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.12.011 * Corresponding author. E-mail addresses: noormaslam@hotmail.com (M.A. Noor), waseemasg@gmail.com (W.A. Khan), dr_akhtar@comsats.edu.pk (A. Hussain). Applied Mathematics and Computation 189 (2007) 1268–1273 www.elsevier.com/locate/amc