Preconditioned eigensolvers - an oxymoron?

Andrew Knyazev

Department of Mathematics, University of Colorado at Denver
P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364.


We attempt to give a formal definition of preconditioned eigensolvers as polynomial methods. We present a survey of some results, mostly theoretical convergence rate estimates, for preconditioned iterative methods for symmetric eigenvalue problems. We consider preconditioned analogs of the power method, the steepest descent/ascent method, the Lanczos-type methods by Scott and Davidson, the conjugate gradient methods, as well as their block variants. We discuss possible approaches for deriving formulas of the methods and conclude that different approaches lead to the same methods. We argue against a popular choice of constructing a preconditioner for the shifted matrix that appeared in the Rayleigh Quotient Method.