IMPROVEMENTS OF THE LOCAL CONVERGENCE OF NEWTON'S METHOD WITH FOURTH-ORDER CONVERGENCE
IOANNIS K. ARGYROS
Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
SANTHOSH GEORGE *
Department of Mathematical and Computational Sciences, NIT Karnataka, 575 025, India
*Author to whom correspondence should be addressed.
Abstract
We present a local convergence analysis of a two-step Newton method in order to approximate a locally unique solution of a nonlinear equation. In earlier studies such as [9, 10] the convergence order of these methods was given under hypotheses reaching up to the third derivative of the function although only the rst derivative appears in these methods. The originality of this paper is twofold. On the one hand, we expand the applicability of these methods by showing convergence using only the first derivative. On the other hand, we compare the convergence radii and provide computable error estimates using only Lipschitz constants, which has not been done before for these methods.
Keywords: Traub-Ostrowski method, fourth order method, local convergence