Convergence Rates for Newton's Method at Singular Points

Creators: Decker, D. W.; Keller, H. B.; Kelley, C. T.

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, the convergence of the Newton iterates to the root is linear rather than quadratic. In this paper we give a detailed analysis of the linear convergence rates for several types of singular problems. For some of these problems we describe modifications of Newton's method which will restore quadratic convergence.

Additional Information

© 1983 Society for Industrial and Applied Mathematics. Received by the editors February 22, 1982, and in revised form May 25, 1982. The research of this author was supported by the National Science Foundation under grant MCS-81-04254. The research of this author was supported by the Army Research Office under Contract DAAG 29-78-C-0011 and by the Department of Energy under Contract EX-76-S-03-0767 Project Agreement No. 12. The research of this author was supported by the National Science Foundation under grant MCS-7902659A01. The authors would like to thank Professor A. O. Griewank of Southern Methodist University and the referee for some very useful comments on the original version of this paper.

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