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

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