<title>Matrix Triangularization By Systolic Arrays</title>

Abstract

Given an n x p matrix X with p < n, matrix triangularization, or triangularization in short, is to determine an n x n nonsingular matrix Al such that MX = [ R 0 where R is p x p upper triangular, and furthermore to compute the entries in R. By triangularization, many matrix problems are reduced to the simpler problem of solving triangular- linear systems (see for example, Stewart). When X is a square matrix, triangularization is the major step in almost all direct methods for solving general linear systems. When M is restricted to be an orthogonal matrix Q, triangularization is also the key step in computing least squares solutions by the QR decomposition, and in computing eigenvalues by the QR algorithm. Triangularization is computationally expensive, however. Algorithms for performing it typically require n3 operations on general n x n matrices. As a result, triangularization has become a bottleneck in some real-time applications.11 This paper sketches unified concepts of using systolic arrays to perform real-time triangularization for both general and band matrices. (Examples and general discussions of systolic architectures can be found in other papers.^6.7) Under the same framework systolic triangularization arrays arc derived for the solution of linear systems with pivoting and for least squares computations. More detailed descriptions of the suggested systolic arrays will appear in the final version of the paper.