A Block Fourier Decomposition Method

Abstract

In this paper, we present a parallel block decomposition method for a special class of block tridiagonal matrices of the form K = block-tridiag [B, A, B] and its variants, where A and B, A,B, ∈ , are general square matrices. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis (simply referred to as FPS in this paper). Unlike FPS, our approach orthogonally decomposes K into q diagonal blocks, instead of p blocks as achieved by FPS. Furthermore, FPS employs the eigenpairs of A and B and requires that A and B be symmetric and commute. Our approach does not impose any such constraints and uses only the eigenvectors of the tridiagonal matrix T = tridiag [b, a, b] and its variants, in a block form, where a and b are scalars. Therefore, this approach has wider applications than FPS and lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems as well.

This work was supported in part by the Army Research Laboratory under Grant No. DAAL01-98-2-D065.