Solving sparse linear least-squares problems on some supercomputers by using large dense blocks

Efficient subroutines for dense matrix computations have recently been developed and are available on many high-speed computers. On some computers the speed of many dense matrix operations is near to the peak-performance. For sparse matrices storage and operations can be saved by operating only and storing only nonzero elements. However, the price is a great degradation of the speed of computations on supercomputers (due to the use of indirect addresses, to the need to insert new nonzeros in the sparse storage scheme, to the lack of data locality, etc.).On many high-speed computers a dense matrix technique is preferable to sparse matrix technique when the matrices are not large, because the high computational speed compensates fully the disadvantages of using more arithmetic operations and more storage. For very large matrices the computations must be organized as a sequence of tasks in each of which a dense block is treated. The blocks must be large enough to achieve a high computational speed, but not too large, because this will lead to a large increase in both the computing time and the storage. A special “locally optimized reordering algorithm” (LORA) is described, which reorders the matrix so that dense blocks can be constructed and treated with some standard software, say LAPACK or NAG.These ideas are implemented for linear least-squares problems. The rectangular matrices (that appear in such problems) are decomposed by an orthogonal method. Results obtained on a CRAY C92A computer demonstrate the efficiency of using large dense blocks.

[1]  Ole Østerby,et al.  Direct Methods for Space Matrices , 1983, Lecture Notes in Computer Science.

[2]  George Ostrouchov.  Symbolic givens reduction and row-ordering in large sparse least squares problems , 1987 .

[3]  W. Morven Gentleman,et al.  Row elimination for solving sparse linear systems and least squares problems , 1976 .

[4]  Iain S. Duff,et al.  Full matrix techniques in sparse Gaussian elimination , 1982 .

[5]  Zahari Zlatev,et al.  Parallel Solution of Sparse Problems by Using a Sequence of Large Dense Blocks , 1996, PARA.

[6]  Deborah Sulsky,et al.  Row Ordering for a Sparse QR Decomposition , 1994 .

[7]  Z. Zlatev.  Computational Methods for General Sparse Matrices , 1991 .

[8]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[9]  Pontus Matstoms,et al.  Parallel Sparse QR Factorization on Shared Memory Architectures , 1995, Parallel Comput..

[10]  Zahari Zlatev,et al.  A Parallel Sparse QR-Factorization Algorithm , 1995, PARA.

[11]  W. E. Gentleman.  Least Squares Computations by Givens Transformations Without Square Roots , 1973 .

[12]  John G. Lewis,et al.  Sparse matrix test problems , 1982, SGNM.

[13]  P. Hansen,et al.  Improving the numerical stability and the performance of a parallel sparse solver , 1995 .

[14]  I. Duff,et al.  Multifrontal QR Factorization in a Multiprocessor Environment , 1996 .

[15]  O. Østerby,et al.  Direct Methods for Sparse Matrices , 1983 .

[16]  Iain S. Duff.  A review of frontal methods for solving linear systems , 1996 .

[17]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[18]  Zahari Zlatev,et al.  A testing scheme for subroutines solving large linear problems , 1981, Comput. Chem..

[19]  John K. Reid,et al.  Some Design Features of a Sparse Matrix Code , 1979, TOMS.

[20]  Z. Zlatev.  Comparison of two pivotal strategies in sparse plane rotations , 1980 .

[21]  A. George,et al.  Row-ordering schemes for sparse givens transformations. II. implicit graph model , 1984 .

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

[23]  Alan George,et al.  Row-ordering schemes for sparse givens transformation. III. analyses for a model problem☆ , 1986 .

[24]  John K. Reid,et al.  On applying Householder transformations to linear least squares problems , 1968, IFIP Congress.

[25]  John K. Reid,et al.  A Note on the Least Squares Solution of a Band System of Linear Equations by Householder Reductions , 1967, Comput. J..