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

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.