Modified Gram-Schmidt QR Factorization on Hypercube SIMD Computers

Abstract QR factorization is a popular calculation method in matrix algebra due to its usefulness in the solution of problems such as estimating least squares and calculating eigenvalues. In this paper, we describe a parallel algorithm for the calculation of the QR factorization on a hypercube architecture of the SIMD type with distributed memory. We have chosen the modified Gram-Schmidt method with pivoting to determine the QR factorization as it is characterized by good numerical stability. As an application of the QR factorization, we analyze the problem of least squares, developing a complementary parallel algorithm for solving it. Both algorithms are general; they are not limited by the size of the problem or the dimension of the hypercube. Finally, we analyze the algorithmic complexities of both parallel algorithms.

[1]  Yves Robert,et al.  Systolic Givens factorization of dense rectangular matrices , 1988 .

[2]  Javier D. Bruguera,et al.  ACLE: A Software Package for SIMD Computer Simulation , 1990, Comput. J..

[3]  M. J. D. Powell,et al.  QR Factorization for Linear Least-Squares Problems on a Hypercube Multiprocessor , 1988 .

[4]  Charles L. Seitz,et al.  The cosmic cube , 1985, CACM.

[5]  Juan José Navarro Guerrero,et al.  A one dimensional systolic array to solve the QR iteration for matrices of any size , 1985 .

[6]  Javier D. Bruguera,et al.  Gaussian elimination with pivoting on hypercubes , 1990, Parallel Comput..

[7]  Francisco F. Rivera,et al.  Parallel Squared Error Clustering on Hypercube Arrays , 1990, J. Parallel Distributed Comput..

[8]  Christian H. Bischof.  A parallel QR factorization algorithm using local pivoting , 1988, Proceedings. SUPERCOMPUTING '88.

[9]  David A. Padua,et al.  Utilizing Multidimensional Loop Parallelism on Large-Scale Parallel Processor Systems , 1989, IEEE Trans. Computers.

[10]  Javier D. Bruguera,et al.  Software tools for multiprocessor simulation and programming , 1990 .

[11]  Daniel Boley,et al.  A parallel QR algorithm for the nonsymmetric eigenvalue problem , 1989 .

[12]  Kon Max Wong,et al.  A Fast QR-Based Array-Processing Algorithm , 1988, Optics & Photonics.

[13]  C. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[14]  R. M. Chamberlain,et al.  Gray codes, Fast Fourier Transforms and hypercubes , 1988, Parallel Comput..

[15]  Francisco F. Rivera,et al.  Parallel fuzzy clustering on fixed size hypercube SIMD computers , 1989, Parallel Comput..

[16]  Javier D. Bruguera,et al.  A parallel markovian model reliability algorithm for hypercube networks , 1989 .

[17]  Ming-Syan Chen,et al.  Processor Allocation in an N-Cube Multiprocessor Using Gray Codes , 1987, IEEE Transactions on Computers.

[18]  Fuyun Ling,et al.  A recursive modified Gram-Schmidt algorithm for least- squares estimation , 1986, IEEE Trans. Acoust. Speech Signal Process..

[19]  G. C. Fox,et al.  Solving Problems on Concurrent Processors , 1988 .

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

[21]  Javier D. Bruguera,et al.  Image reconstruction on hypercube computers , 1990, [1990 Proceedings] The Third Symposium on the Frontiers of Massively Parallel Computation.

[22]  John P. Hayes,et al.  A Microprocessor-based Hypercube Supercomputer , 1986, IEEE Micro.

[23]  G. W. Stewart,et al.  A parallel implementation of the QR-algorithm , 1987, Parallel Comput..

[24]  Tariq S. Durrani,et al.  Arithmetic implementation of the Givens QR triarray , 1989, International Conference on Acoustics, Speech, and Signal Processing,.

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

[26]  David S. Watkins,et al.  Understanding the $QR$ Algorithm , 1982 .

[27]  Udo Schendel,et al.  Introduction to Numerical Methods for Parallel Computers , 1984 .

[28]  Sukil Kim,et al.  Least-Squares Multiple Updating Algorithms on a Hypercube , 1990, J. Parallel Distributed Comput..