A Subspace Preconditioning Algorithm
for Eigenvector/Eigenvalue Computation

J. H. Bramble
Department of Mathematics
Cornell University
Ithaca, NY 14853

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

J. E. Pasciak
Applied Math Department
Brookhaven National Laboratory
Upton, NY 11973


We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V. In our applications, the dimension of V is large and the cost of inverting A is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning for A. Estimates will be provided which show that the preconditioned method converges linearly when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.

Contributed November 6, 1995.