ar X iv : 0 80 1 . 30 99 v 2 [ m at h . N A ] 1 6 Ju n 20 08 GRADIENT FLOW

Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but their convergence theory remains sparse and complex. We consider the simplest preconditioned eigensolver— the gradient iterative method with a fixed step size—for symmetric generalized eigenvalue problems, where we use the gradient of the Rayleigh quotient as an optimization direction. We prove a known sharp and simple convergence rate bound using novel geometric ideas. The proof is short and selfcontained. We first demonstrate that, for a given initial eigenvector approximation, the next iterative approximation belongs to a cone if we apply all admissible preconditioners. Second, using the classical Temple inequality we easily determine that a subspace spanned by two specific eigenvectors gives the smallest norm of the gradient of the Rayleigh quotient. Then we analyse a corresponding continuation method of a gradient flow of the Rayleigh quotient and show that the slowest convergence of the continuation method is reached when the initial vector belongs to this two-dimensional subspace. Next, we extend this result by integration to our fixed step gradient method to conclude that the point on the cone, which corresponds to the poorest convergence and thus gives the guaranteed convergence rate bound, belongs to the same two-dimensional invariant subspace. This reduces the convergence analysis to a two-dimensional case, where a sharp convergence rate bound is derived.