Rates of Convergence of Newton's Method

摘要

Given an operator P in a Banach space X with Lipschitz continuous derivative P primed, it is shown that the existence of 1/(P primed (x + 1)) is necessary and sufficient to predict on the basis of the theorem of L. V. Kantorovic that the Newton sequence x sub (n + 1) = (x sub n) - P(x sub n)/(P primed(s sub n)) will converge to a solution x of the equation P(x) = o quadratically. Some examples are given of convergent Newton sequences for which convergence and the rate of convergence cannot be predicted by the Kantorovic theorem. (Author)