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Computing Row and Column Counts for Sparse QR and LU Factorization

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Abstract

We present algorithms to determine the number of nonzeros in each row and column of the factors of a sparse matrix, for both the QR factorization and the LU factorization with partial pivoting. The algorithms use only the nonzero structure of the input matrix, and run in time nearly linear in the number of nonzeros in that matrix. They may be used to set up data structures or schedule parallel operations in advance of the numerical factorization.

The row and column counts we compute are upper bounds on the actual counts. If the input matrix is strong Hall and there is no coincidental numerical cancellation, the counts are exact for QR factorization and are the tightest bounds possible for LU factorization.

These algorithms are based on our earlier work on computing row and column counts for sparse Cholesky factorization, plus an efficient method to compute the column elimination tree of a sparse matrix without explicitly forming the product of the matrix and its transpose.

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Authors and Affiliations

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

    J. R. Gilbert

  2. National Energy Research Scientific Computing Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS 50F, Berkeley, CA, 94720, USA

    X. S. Li & E. G. Ng

  3. Computer Science and Mathematics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN, 37831-6367, USA

    B. W. Peyton

Authors
  1. J. R. Gilbert
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  2. X. S. Li
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  3. E. G. Ng
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  4. B. W. Peyton
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Gilbert, J.R., Li, X.S., Ng, E.G. et al. Computing Row and Column Counts for Sparse QR and LU Factorization. BIT Numerical Mathematics 41, 693–710 (2001). https://doi.org/10.1023/A:1021943902025

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