IMPROVEMENTS OF THE LOCAL CONVERGENCE OF NEWTON'S METHOD WITH FOURTH-ORDER CONVERGENCE

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.