A NEW PROOF OF SUPERLINEAR CONVERGENCE FOR BROYDEN'S METHOD IN HILBERT SPACE

Broyden's method is an extension of the secant method for an equation in one real variable to an arbitrary Hilbert space setting. It is a result of Griewank that the Broyden iterates converge locally superlinearly to a root if, in addition to the assumptions needed in finite dimension, the initial approximation for the Frechet derivative differs from the Frechet derivative at the root only by a compact operator. In this paper a new and much simpler proof of this theorem is given based on the concept of collective compactness.