On Parallelizable Eigensolvers

In this paper, we show how the intimate relationship between a certain matrix algebra and the ring of complex polynomials suggests a strategy for devising new parallel algorithms for the eigenproblem with a twofold potential for parallelism. Speciically, we explore the algorithmic implications of the fact that an ability to compute nontrivial invariant subspaces of a matrix M results in an ability to perform block triangularization of M into two independent subproblems. We further propose a strategy for devising algorithms to compute nontrivial invariant subspaces of M which rely almost exclusively on two computational primitives: matrix multiplication and solving systems of linear equations. The future success of numerical computation must certainly depend on the availability of algorithms to perform these primitives fast on parallel architectures. The recursive application of the block trian-gularization scheme on the subproblems generated leads naturally to a divide and conquer strategy.