NUMERICALLY STABLE SOLUTION OF DENSE SYSTEMS OF LINEAR EQUATIONS USING MESH-CONNECTED PROCESSORS

A. BOJANCZYK, R. P. BRENT, AND H. T. KUNG

Abstract

We propose a multiprocessor structure for solving a dense system of n linear equations. The solution is obtained in two stages. First, the matrix of coefficients is reduced to upper triangular form via Givens rotations. Second, a back substitution process is applied to the triangular system. A two-dimensional array of Θ(n^{2}) processors is employed to implement the first stage, and (using a previously known scheme) a one-dimensional array of Θ(n) processors is employed to implement the second stage. These processor arrays allow both stages to be carried out in time O(n), and they are well suited for VLSI implementation as identical processors with a simple and regular interconnection pattern are required.

Comments

Only the Abstract is given here. The full paper appeared as [1]. Although publication of [1] was delayed, it predated the well-known work of Gentleman and Kung [3]. As shown in [2], the algorithms of [1] and [3] are in a certain sense dual. The algorithm of [3] is preferable for QR factorization of m × n matrices where m > n (as in least squares problems) because the size of the processor array depends on n but not on m. For later work, see [2, 4].

References

1. A. W. Bojanczyk, R. P. Brent and H. T. Kung, “Numerically stable solution of dense systems of linear equations using mesh-connected processors”, SIAM J. Scientific and Statistical Computing 5 (1984), 95–104. MR 85b:65026. Also appeared as Report TR-CS-81-01, Department of Computer Science, ANU; and as Report TR-CS-81-119, Department of Computer Science, CMU (May 1981), 23 pp. rpb065.

2. R. P. Brent, “Parallel algorithms for digital signal processing”, Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms (edited by G. H. Golub and P. Van Dooren), Springer-Verlag, 1991, 93–110. rpb110.

3. W. M. Gentleman and H. T. Kung, “Matrix triangularization by systolic arrays”, Proc. SPIE, Volume 298, Real-Time Signal Processing IV, SPIE, August 1981.

1991 Mathematics Subject Classification. Primary 65Y05; Secondary 65F05, 65F25, 65Y10, 68Q35. Key words and phrases. Givens method, least squares, linear systems, numerical stability, orthogonal

factorization, parallel algorithms, QR method, special-purpose hardware, systolic arrays, VLSI. Received by the editors June 10, 1981, and in revised form July 15, 1982. The research of H. T. Kung was supported in part by the National Science Foundation under grant

MCS78-236-76.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1984, SIAM. Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb065a typeset using AMS-L^{A}TEX 4. F. T. Luk, “A rotation method for computing the QR-decomposition”, SIAM J. Scientific and Statis-

tical Computing 7 (1986), 452–459.

(Bojanczyk) Institute for Informatics, University of Warsaw, PKiN, p. 850, 00-901 Warsaw, Poland

(Brent) Department of Computer Science, Australian National University, Canberra, Australia

(Kung) Department of Computer Science, Carnegie-Mellon University, Pittsburgh, PA 15213, USA