Convergence of Broyden’s Method in Banach Spaces

Abstract

This paper proves new convergence theorems for convergence of Broyden’s method when applied to nonlinear equations in Banach spaces. The convergence is in the norm of the Banach space itself, rather than in the norm of some Hilbert space that contains the Banach space. It is shown that the norms in which q-superlinear convergence takes place are determined by the smoothing properties of the error in the Fréchet derivative approximation and not by the inner product in which Broyden’s method is implemented. Among the consequences of the results in this paper are a proof of sup-norm local q-superlinear convergence when Broyden’s method is applied to integral equations with continuous kernels, global q-superlinear convergence of the Broyden iterates for singular and nonsingular linear compact fixed point problems in Banach space, a new method for integral equations having derivatives with sparse kernels, and q-superlinear convergence for a new method for integral equations when part of the Fréchet derivative can be explicitly computed. Partitioned variants of the methods and the “bad” Broyden method are also discussed.