James H. Bramble, Andrew V. Knyazev (speaker), and Joseph E. Pasciak
A Subspace Preconditioning Algorithm
for Eigenvector/Eigenvalue Computation
We consider the problem of computing a modest number
of the smallest eigenvalues along with orthogonal bases for the
corresponding eigen-spaces of a symmetric positive definite
matrix. In our applications, the
dimension of a matrix is large and the cost of its
inverting is prohibitive.
In this paper, we shall develop an effective
parallelizable technique for
computing these eigenvalues and eigenvectors utilizing subspace
iteration and preconditioning.
Estimates will be provided which
show that the preconditioned method converges linearly and uniformly
in the matrix dimension when used with a uniform
preconditioner under the assumption that the approximating subspace is
close enough to the span of desired eigenvectors.