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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 687382, 7 pages
http://dx.doi.org/10.1155/2013/687382
Research Article

A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence

Department of Environmental Health Science, University of Swaziland, P.O. Box 369, Mbabane H100, Swaziland

Received 25 October 2012; Revised 25 December 2012; Accepted 17 February 2013

Academic Editor: Michele Benzi

Copyright © 2013 Ababu Teklemariam Tiruneh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method.