Simultaneous point estimates for Newton's method

Abstract

Beside the classical Kantorovich theory there exist convergence criteria for the Newton iteration which only involve data at one point, i.e. point estimates. Given a polynomial P, these conditions imply the point evaluation of n = deg(P) functions (from a certain Taylor expansion). Such sufficient conditions ensure quadratic convergence to a single zero and have been used by several authors in the design and analysis of robust, fast and efficient root-finding methods for polynomials. In this paper a sufficient condition for the simultaneous convergence of the one-dimensional Newton iteration for polynomials will be given. The new condition involves only n point evaluations of the Newton correction and the minimum mutual distance of approximations to ensure "simultaneous" quadratic convergence to the pairwise distinct n roots.