On the convergence of a Newton-like method in ℝn and the use of Berinde's exit criterion

Sen, Rabindranath; Mukherjee, Sulekha; Patra, Rajesh

doi:10.1155/IJMMS/2006/36482

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International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 36482, 9 pages
http://dx.doi.org/10.1155/IJMMS/2006/36482

On the convergence of a Newton-like method in ℝn and the use of Berinde's exit criterion

Rabindranath Sen,¹ Sulekha Mukherjee,² and Rajesh Patra²

¹Department of Applied Mathematics, University College of Science, 92 A.P.C. Road, Calcutta 700009, India
²Department of Mathematics, University of Kalyani, Kalyani 741 235, West Bengal, India

Received 1 January 2006; Revised 11 August 2006; Accepted 21 August 2006

Copyright © 2006 Hindawi Publishing Corporation. 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

Berinde has shown that Newton's method for a scalar equation f(x)=0 converges under some conditions involving only f and f′ and not f″ when a generalized stopping inequality is valid. Later Sen et al. have extended Berinde's theorem to the case where the condition that f′(x)≠0 need not necessarily be true. In this paper we have extended Berinde's theorem to the class of n-dimensional equations, F(x)=0, where F:ℝn→ℝn, ℝn denotes the n-dimensional Euclidean space. We have also assumed that F′(x) has an inverse not necessarily at every point in the domain of definition of F.