A geometric theory for preconditioned inverse iteration. III: A short and sharp convergence estimate for generalized eigenvalue problems

In two previous papers by Neymeyr: A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient, LAA 322: (1-3), 61-85, 2001, and A geometric theory for preconditioned inverse iteration II: Convergence estimates, LAA 322: (1-3), 87-104, 2001, a sharp, but cumbersome, convergence rate estimate was proved for a simple preconditioned eigensolver, which computes the smallest eigenvalue together with the corresponding eigenvector of a symmetric positive definite matrix, using a preconditioned gradient minimization of the Rayleigh quotient. In the present paper, we discover and prove a much shorter and more elegant, but still sharp in decisive quantities, convergence rate estimate of the same method that also holds for a generalized symmetric definite eigenvalue problem. The new estimate is simple enough to stimulate a search for a more straightforward proof technique that could be helpful to investigate such practically important method as the locally optimal block preconditioned conjugate gradient eigensolver. We demonstrate practical effectiveness of the latter for a model problem, where it compares favorably with two well-known Jacobi-Davidson type methods, JDQR and JDCG.