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

In two previous papers by Neymeyr [Linear Algebra Appl. 322 (1–3) (2001) 61; 322 (1–3) (2001) 87], 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 a practically important method as the locally optimal block preconditioned conjugate gradient eigensolver.