Diffeomorphisms and newton-direction algorithms

This paper shows that an iterative process, with certain desirable convergence properties, can be used to compute the solution of an important general equation when certain conditions are met. More specifically, let f be a function from U into B, where B is a Banach space and U is a nonempty open subset of B. One main result reported on is a proof of the existence of a superlinearly convergent algorithm that globally converges to a solution x of f(x) = a for each a &sin; B, whenever f is a C
¹
-diffeomorphism of U onto B, and either B = R
ⁿ
or f satisfies certain other conditions that are frequently met in applications. For the case of an important class of monotone diffeomorphisms f in a Hilbert space H (examples arise, for example, in signal-theory studies), the “other conditions” reduce to simply the requirement that f' (the F-derivative of f) be uniformly continuous on closed bounded subsets of H.