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

A modification of the iterative matrix diagonalization method of Davidson is presented that is applicable to the symmetric eigenvalue problem. This 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 eigenpairs by reducing the number of matrix–vector products that must be computed with the exact matrix. Several applications are presented. These are chosen to show the range of applicability of the method, the convergence behavior for a wide range of matrix types, and also the wide range of approaches that may be employed to generate approximate matrices.