International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 36482, 9 pages
doi:10.1155/IJMMS/2006/36482
On the convergence of a Newton-like method in ℝn and the use of Berinde's exit criterion
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 Rabindranath Sen 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
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.