The Graves Theorem Revisited II: Robust Convergence of the Newton Method

AMS subject classification: 65J15, 47H04, 90C30.

Based on the original proof of the Graves theorem [9] we study the convergence of the Newton method for the solution of the equation f (x) = y, uniform with respect to the starting point and the parameter y. We show that the surjectivity of the Jacobian implies the Aubin continuity, relative to the supremum norm, of the map taking the starting point and the parameter y to the set of all Newton sequences. These results complement our previous paper [4].

This work was supported by The National Science Foundation. The revised version of this paper was prepared during author’s visit at the University of Zürich, Switzerland.