RECURRENCE RELATIONS FOR RATIONAL CUBIC METHODS .1. THE HALLEY METHOD

In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of "recurrence relations", analogous to those given for the Newton method by Kantorovich, improving previous results by Döring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper. © 1990 Springer-Verlag.