We have developed an approach to extending iterative eigenvalue methods based upon the use of approximating matrices. This approach is most useful in the case where the calculation of matrix-vector products for the eigenvalue matrix is expensive and where a sequence of approximating matrices can be used which involve inexpensive matrix-vector products. Our particular focus has been on the extension of Davidson's method as applied to this class of numerical problems.
Specifically, this approach is a modification of the iterative matrix diagonalization method of Davidison and is applicable to symmetric eigenvalue problems. The method is based on subspace projections of a sequence of one or more approximate matrices. The purpose of these approximate matrices is to improve the efficiency of the solution of the desired eigenvpairs by reducing the number of matrix-vector products that must be computed with the exact matrix. Several applications have been studied, which have been chosen to show the range of applicability of the method, the convergence behavior for a wide range of matrix types, and the wide range of approaches that may be employed to generate these approximating matrices.
Initial results of this approach appear in
R. Shepard, A. F. Wagner, J. L. Tilson, and M. Minkoff, "The Subspace
Projected Approximate Matrix (SPAM) Modification of the Davidson Method," J.
Computational Physics 172(2) (2001) 472-514. Also Preprint
ANL/MCS-P878-0401, April 2001. SPAM (Subspace Projected
Approximate Matrices)
The software is available at
README documentation
SPAM software