Diffeomorphisms and Newton-Direction Algorithms

Diffeomorphisms and Newton-Direction Algorithms* By I. W. SANDBERG (Manuscript received March 26, 1980) 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 ofB. 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 G B, whenever f is a C1 -diffeomorphism of U onto B, and either B -- Rn 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 signaltheory studies), the "other conditions" reduce to simply the requirement that f'{the F-derivative o f f ) be uniformly continuous on closed bounded subsets of H. I. INTRODUCTION AND OUTLINE OF RESULTS Let f be a function from U into B, where B is a Banach space with norm | · | and U is a nonempty open subset of B. We say that f is differentiable on a set S C U if /has a Frechet derivative f'(s) at each point s of S.f (If, for example, B = Rn with the usual Euclidean norm, then f is differentiable on U if it is continuously differentiable on U in the usual sense.) By f a C1 -diffeomorphism, we mean that f is a homeomorphism of U onto B, and f' and ( f _ 1 ) ' exist and are continuous on U and B, respectively.