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Predicting Structure in Nonsymmetric Sparse Matrix Factorizations

  • Conference paper
Graph Theory and Sparse Matrix Computation

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 56))

Abstract

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.

Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, California 94304–1314 (gilbert@parc.xerox.com). This work was supported in part by the Christian Michelsen Institute, Bergen, Norway, and by the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation. Copyright © 1992 by Xerox Corporation. All rights reserved.

Mathematical Sciences Section, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831–6367 (ngeg@ornl.gov). This author’s work was supported by the Applied Mathematical Sciences Research Program of the Office of Energy Research, U.S. Department of Energy, under contract DE-ACO5–840R21400 and by the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation.

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Author information

Authors and Affiliations

  1. Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, California, 94304-1314, USA

    John R. Gilbert

  2. Mathematical Sciences Section, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee, 37831-6367, USA

    Esmond G. Ng

Authors
  1. John R. Gilbert
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  2. Esmond G. Ng
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Editor information

Editors and Affiliations

  1. University of Waterloo, Needles Hall, Waterloo, Ontario, N2L 3G1, Canada

    Alan George

  2. Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, CA, 94304-1314, USA

    John R. Gilbert

  3. Department of Computer Science, York University, North York, Ontario, M3J 1P3, Canada

    Joseph W. H. Liu

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© 1993 Springer-Verlag New York, Inc.

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Gilbert, J.R., Ng, E.G. (1993). Predicting Structure in Nonsymmetric Sparse Matrix Factorizations. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds) Graph Theory and Sparse Matrix Computation. The IMA Volumes in Mathematics and its Applications, vol 56. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8369-7_6

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  • DOI: https://doi.org/10.1007/978-1-4613-8369-7_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8371-0

  • Online ISBN: 978-1-4613-8369-7

  • eBook Packages: Springer Book Archive

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