Parallel computation of orthogonal factors of sparse matrices
- Pennsylvania State Univ., University Park, PA (United States)
We study the solution at the least squares problem for a large and sparse overdetermined matrix A by QR factorization A and transformation the right-hand side vector b to Q{sup T}b. An efFicient method for computing Q{sup T}b on P-processor distributed memory machines is presented. In this paper, we determined the number of multiplications required to compute Q{sup T}b using the multifrontal Householder QR factorization by Lewis, Pierce, and Wah[Technical report ECA-TR-127, Boeing computer services(1989)]. Instead of storing the Q matrix itself, we store all the nonzero parts of the Householder transformation matrices Qi`s of all the multifrontal matrices Fi`s of A. We also use Schreiber and Van Loan`s Storage-Efficient-WY Representation [SIAM J. Sci. Stat. Computing, 19(1989), pp 55-57] for each Qi to introduce BLAS-2 operations. A theoretical operation count for the K by K unbordered grid model problem shows that the proposed method requires O(N{sub R}) multiplications (O(K{sup 2}log K) for the K by K grid model problems) to compute Q{sup T}b, where N{sub R} represents the number of nonzeros of the Cholesky factor of A. Some numerical results for the grid model problems as well as Harwell-Boeing problems on both sequential machines and iPSC distributed memory machines are provided.
- OSTI ID:
- 54445
- Report Number(s):
- DOE/ER/25151-1-Vol.1; CONF-930331-Vol.1; CNN: Grant CCR-9201692; TRN: 94:007584-0085
- Resource Relation:
- Conference: 6. Society for Industrial and Applied Mathematics (SIAM) conference on parallel processing for scientific computing, Norfolk, VA (United States), 21-24 Mar 1993; Other Information: PBD: 1993; Related Information: Is Part Of Parallel processing for scientific computing: Proceedings. Volume 1; Sincovec, R.F.; Leuze, M.R. [eds.] [Oak Ridge National Lab., TN (United States)]; Keyes, D.E. [ed.] [Yale Univ., New Haven, CT (United States)]; Petzold, L.R. [ed.] [Minnesota Univ., Minneapolis, MN (United States)]; Reed, D.A. [ed.] [Illinois Univ., Chicago, IL (United States)]; PB: 522 p.
- Country of Publication:
- United States
- Language:
- English
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