Structure-preserving and rank-revealing QR-factorizations

Journal Article · Fri Nov 01 00:00:00 EST 1991 · SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States)

DOI:https://doi.org/10.1137/0912073· OSTI ID:7195441

Bischof, C H ^[1]; Hansen, P C ^[2]

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1. Argonne National Labs., IL (United States). Mathematics and Computer Science Div.
2. Technical Univ. of Denmark, Lyngby (Denmark). Danish Computing Center for Research and Education

The rank-revealing QR-factorization (RRQR-factorization) is a special QR-factorization that is guaranteed to reveal the numerical rank of the matrix under consideration. This makes the RRQR-factorization a useful tool in the numerical treatment of many rank-deficient problems in numerical linear algebra. In this paper, a framework is presented for the efficient implementation of RRQR algorithms, in particular, for sparse matrices. A sparse RRQR-algorithm should seek to preserve the structure and sparsity of the matrix as much as possible while retaining the ability to capture safely the numerical rank. To this end, the paper proposes to compute an initial QR-factorization using a restricted pivoting strategy guarded by incremental condition estimation (ICE), and then applies the algorithm suggested by Chan and Foster to this QR-factorization. The column exchange strategy used in the initial QR factorization will exploit the fact that certain column exchanges do not change the sparsity structure, and compute a sparse QR-factorization that is a good approximation of the sought-after RRQR-factorization. Due to quantities produced by ICE, the Chan/Foster RRQR algorithm can be implemented very cheaply, thus verifying that the sought-after RRQR-factorization has indeed been computed. Experimental results on a model problem show that the initial QR-factorization is indeed very likely to produce RRQR-factorization.

• DOE Contract Number: W-31109-ENG-38
• OSTI ID: 7195441
• Journal Information: SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States), Vol. 12:6; ISSN 0196-5204
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
FACTORIZATION
ALGORITHMS
EXPERIMENTAL DATA
LEAST SQUARE FIT
MATRICES
NUMERICAL SOLUTION
PARALLEL PROCESSING
PERFORMANCE
DATA
INFORMATION
MATHEMATICAL LOGIC
MAXIMUM-LIKELIHOOD FIT
NUMERICAL DATA
PROGRAMMING
990200* - Mathematics & Computers