Stabilized bordered block diagonal forms for parallel sparse solvers

One possible approach to the solution of large sparse linear systems is to reorder the system matrix to bordered block diagonal form and then to solve the block system in parallel. We consider the duality between singly bordered and doubly bordered block diagonal forms. The idea of a stabilized doubly bordered block diagonal form is introduced. We show how a stable factorization of a singly bordered block diagonal matrix results in a stabilized doubly bordered block diagonal matrix. We propose using matrix stretching to generate a singly bordered form from a doubly bordered form. Matrix stretching is compared with two alternative methods for obtaining a singly bordered form and is shown to be efficient both in computation time and the quality of the resulting block structure.