An algorithm and architecture for the parallel solution of systems of linear equations

Abstract

This paper evaluates a paradigm for the efficient utilization of commercially available processors to implement serial algorithms on a parallel architecture. We present an architecture based on this paradigm as well as an algorithm for the parallel solution of a nonhomogeneous system of linear equations with constant coefficients. Major advantages stem from its systolic-like array structure and the versatility of fully programmable processor elements. The method uses a Givens rotation implementation of the well known QR factorization. Unlike other direct methods of factorization followed by backsubstitution, this implementation of the algorithm avoids the backsubstitution bottleneck. The computational complexity of this feedforward direct method of solving nonsingular systems of linear equations is on the order that of QR matrix factorization. Due to the programmability of the processor in the array, the mapping of this algorithm extends to an entire family of algorithms. We map this family of algorithms onto the novel architecture and present a comprehensive performance analysis. Performance results identify the algorithm/architecture combination as a cost effective, efficient method which exhibits speedup that is directly proportional to the number of processors used.