A Subspace Preconditioning Algorithm
                    for Eigenvector/Eigenvalue Computation

          James H. Bramble, Andrew V. Knyazev, and Joseph E. Pasciak

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.