Predicting structure in nonsymmetric sparse matrix factorizations

Technical Report · Thu Oct 01 04:00:00 EDT 1992

DOI:https://doi.org/10.2172/6987948· OSTI ID:6987948

Gilbert, J R ^[1]; Ng, E G ^[2]

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1. Xerox Palo Alto Research Center, CA (United States)
2. Oak Ridge National Lab., TN (United States)

Many computations on sparse matrices have a phase that predicts the nonzero structure of the output, followed by a phase that actually performs the numerical computation. We study structure prediction for computations that involve nonsymmetric row and column permutations and nonsymmetric or non-square matrices. Our tools are bipartite graphs, matchings, and alternating paths. Our main new result concerns LU factorization with partial pivoting. We show that if a square matrix A has the strong Hall property (i.e., is fully indecomposable) then an upper bound due to George and Ng on the nonzero structure of L + U is as tight as possible. To show this, we prove a crucial result about alternating paths in strong Hall graphs. The alternating-paths theorem seems to be of independent interest: it can also be used to prove related results about structure prediction for QR factorization that are due to Coleman, Edenbrandt, Gilbert, Hare, Johnson, Olesky, Pothen, and van den Driessche.

• Research Organization: Oak Ridge National Lab., TN (United States)
• Sponsoring Organization: DOE; NSF; USDOE, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
• DOE Contract Number: AC05-84OR21400
• OSTI ID: 6987948
• Report Number(s): ORNL/TM-12204; ON: DE93003365
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
ASYMMETRY
DIAGRAMS
FACTORIZATION
MATHEMATICAL LOGIC
MATRICES