A Six-order Modification of Newton’s Method For Solving Nonlinear Equations

A new six order variant of Newton’s method has been developed by extending a third order method of Ozban [14] and its convergence properties have been discussed. The order of convergence of the proposed method is six. Starting with a suitably chosen x0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first order derivatives per iteration. This implies that efficiency index of our method is 1.5651. Proposed method is comparable with the methods of Parhi, and Gupta [15] and that of Kou and Li [7]. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev– Halley type methods. The efficiency of the method is tested on a number of numerical examples. It is observed that our method takes less number of iterations than Newton’s method and the other third order variants of Newton’s method. On comparison with the sixth order methods, it behaves either similarly or better for the examples considered. Copyright c © 2011 Yang’s Scientific Research Institute, LLC. All rights reserved.