Convergence Acceleration for Newton’s Method at Singular Points

Abstract

If Newton’s method is employed to find a root of a map from a Banach space into itself and the derivative is singular at that root, linear convergence of the Newton sequence to the root is the best that one can expect. In this paper we give sufficient conditions under which Newton’s method may be modified to produce a sequence $\{ x_n \} $ such that the subsequence $\{ x_{2n} \} $ converges quadratically to the root.