**A Six-order Variant of Newton’s Method for Solving Nonlinear Equations**

Department of Mathematics, Faculty of Science

Banaras Hindu University

Varanasi, India

e-mail: manoj07777@gmail.com

### Received:

Received: 30 October 2009; revised: 05 December 2009; accepted: 10 December 2009; published online: 30 December 2009

### DOI: 10.12921/cmst.2009.15.02.185-193

### OAI: oai:lib.psnc.pl:674

### Abstract:

A new variant of Newton’s method based on contra harmonic mean has been developed 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. The proposed method is comparable with the methods of Parhi, and Gupta [15] and that of Kou and Li [8]. 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 lesser number of iterations than Newton’s method and the other third order variants of Newton´s method. In comparison with the sixth order methods, it behaves either similarly or better for the examples considered.

### Key words:

efficiency index, function evaluations, iteration function, Newton's method, order of convergence

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