Application and Extension of the The Subspace Projected Approximate Matrix (SPAM) Modification of the Davidson Method

Mike Minkoff
Argonne National Laboratory

Ron Shepard
Argonne National Laboratory

Albert Wagner
Argonne National Laboratory


Abstract

We extend the SPAM method to the iterative calculation of eigenvalues for Hermitian matrices. This approach is a modification of the iterative matrix diagonalization method of Davidson that is applicable to symmetric eigenvalue problem. The method is based on subspace projections of a sequence of one or more approximating matrices. These approximate matrices are used to improve the efficiency of the solution of of the eigenpairs by reducing the number of matrix-vector products that must be computed with the exact matrix. We provide a presentation of the approach in terms of theory and performance analysis. We include demonstrations of numerical results including several applications with illustrations of approximating matrices. In particular, the class of approximating matrices is extended to tensor product methods and results are shown treating problems with over a million variables. Applications from computational chemistry are also explored.