Least-Squares Multiple Updating Algorithms on a Hypercube

Abstract Parallel algorithms for multiple updating algorithms in recursive least-squares computations are investigated. Comparisons of updating algorithms by carefully implemented orthogonal Householder and Givens schemes are made on the iPSC hypercube distributed memory multiprocessor system. Overall, the performance of updating by orthogonal Householder reflections using a row-oriented storage scheme seems superior to those of Givens rotations using a row-oriented storage scheme and the greedy Givens sequence on the hypercube, for our application. In particular, the communication complexity is independent of the number of vectors being updated. The methods we describe can also be adapted to the parallel computation of general orthogonal factorizations involved in least-squares problems. We have in mind applications to windowed recursive least-squares filtering schemes for near real-time computations on distributed memory architectures.

[1]  Sumit Roy,et al.  Decentralized structures for parallel Kalman filtering , 1988 .

[2]  J. J. Modi,et al.  An alternative givens ordering , 1984 .

[3]  M. Cosnard,et al.  Parallel QR decomposition of a rectangular matrix , 1986 .

[4]  Richard P. Brent,et al.  A Note on Downdating the Cholesky Factorization , 1987 .

[5]  Alan George,et al.  QR Factorization of a Dense Matrix on a Hypercube Multiprocessor , 1990, SIAM J. Sci. Comput..

[6]  A. George,et al.  Solution of sparse linear least squares problems using givens rotations , 1980 .

[7]  Alan George,et al.  QR factorization of a dense matrix on a shared-memory multiprocessor , 1989, Parallel Comput..

[8]  D. Sorensen,et al.  A pipelined givens method for computing the QR factorization of a sparse matrix , 1986 .

[9]  Michael T. Heath,et al.  Modified cyclic algorithms for solving triangular systems on distributed-memory multiprocessors , 1988 .

[10]  Christian H. Bischof,et al.  The WY representation for products of householder matrices , 1985, PPSC.

[11]  Padma Raghavan,et al.  Distributed Orthogonal Factorization , 1989 .

[12]  D. P. Agrawal.  Advanced computer architecture : tutorial , 1986 .

[13]  Dharma P. Agrawal,et al.  B-HIVE project: present and future , 1990 .

[14]  W. Gentleman.  Error analysis of QR decompositions by Givens transformations , 1975 .

[15]  S. Thomas Alexander,et al.  Adaptive Signal Processing , 1986, Texts and Monographs in Computer Science.

[16]  James M. Ortega,et al.  Parallel solution of triangular systems of equations , 1988, Parallel Comput..

[17]  S. Alexander,et al.  Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing , 1988 .

[18]  R. J. Plemmons,et al.  Cholesky downdating on a hypercube , 1989, C3P.

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .