On Newton-Direction Algorithms and Diffeomorphisms

* This paper was presented at the Fourteenth Asilomar Conference on Circuits, Systems, and Computers (Pacific Grove, California, November 17-19, 1980). f In other words, f is differentiable on S C U if for each s e S, there is a bounded linear map /"'(s):B-> B such that f(s + h) = f(s) + f'(s)h + o(| h |) as | h | 0. 339 evaluated, not to their boundedness as operators, which is assured by definition*) C '-diffeomorphisms frequently arise in applications. The purpose of this paper is to report on results that complement those in Ref. 1 where a constructive proof is given of the existence of a Newton-direction algorithm that, for each a E B, generates a sequence in U which converges globally and superlinearly to a solution x of f(x) = a whenever f is a C'-diffeomorphism of Uonto B and either B = R or /satisfies certain other conditions that are frequently met in applications. (For the case of an important class of monotone diffeomorphisms fin a Hilbert space H, the "other conditions" reduce to simply the requirement that f be uniformly continuous on closed bounded subsets of H. A specific example in which H is infinite dimensional is given in Ref. 1.) The algorithm described in Ref. 1 typically involves the recursive determination of positive scalars yo, yi, (which determine the successive steplengths) such that a certain ratio Rk (y*) (which depends on the £th iterate x ) lies between prescribed bounds for all k = 0, 1, 2, · · ·. While it is proved that the y can be chosen as required, and that y = 1 for all sufficiently large k, the actual determination of the y* in a specific case would ordinarily require the use of a one-dimensional search procedure for a finite (and possibly large) number of values of k.