An analysis of the properties of the variants of Newton’s method with third order convergence D.K.R. Babajee, M.Z. Dauhoo * Department of Mathematics, Faculty of Science, University of Mauritius, Reduit, Mauritius Abstract For the last five years, the variants of the Newton’s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we investigate about the rela- tionship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited Taylor expansion. We also prove that they are different forms of the Halley method and are all con- tractive iterative methods in a common neighbourhood. We extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Variants of Newton’s method; Non-linear equations; Cubic convergence; Contraction; Multivariate 1. Introduction Iterative methods for finding approximate solutions to the single non-linear equations f(x) = 0 that have no solutions in closed form have various applications in physics and chemistry. Newton’s method [1] is often the method of choice for approximating such solutions. The method is of quadratic convergence at a simple root, that is, the number of good digits is doubled at each iteration. The convergence of the iterates and the rate of convergence play an important role in the design of new iterative methods [2]. There are higher order methods that allow for faster convergence. For instance, Halley’s method [3], Householder’s method [4], dou- ble convex acceleration of Whittaker’s method [5], super-Newton’s method (Chebyshev) [6], super-Halley’s method or the Chebyshev–Halley’s family of iterative methods [7,8] has third order convergence, but require the second derivative which is difficult to evaluate and thus time consuming for complicated functions. To avoid the calculation of second derivatives, some variants of Newton’s method with third order convergence 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.116 * Corresponding author. E-mail addresses: rajiv.nishi@intnet.mu (D.K.R. Babajee), m.dauhoo@uom.ac.mu (M.Z. Dauhoo). Applied Mathematics and Computation 183 (2006) 659–684 www.elsevier.com/locate/amc