Some Convergence Properties of Broyden's Method

Abstract

In 1965 Broyden introduced a family of algorithms called(rank-one) quasi-New-ton methods for iteratively solving systems of nonlinear equations. We show that when any member of this family is applied to an n x n nonsingular system of linear equations and direct-prediction steps are taken every second iteration, then the solution is found in at most 2n steps. Specializing to the particular family member known as Broydenâ€TMs (good) method, we use this result to show that Broyden's method enjoys local 2n-step Q-quadratic convergence on nonlinear problems.