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A block algorithm for computing rank-revealing QR factorizations

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Abstract

We present a block algorithm for computing rank-revealing QR factorizations (RRQR factorizations) of rank-deficient matrices. The algorithm is a block generalization of the RRQR-algorithm of Foster and Chan. While the unblocked algorithm reveals the rank by peeling off small singular values one by one, our algorithm identifies groups of small singular values. In our block algorithm, we use incremental condition estimation to compute approximations to the nullvectors of the matrix. By applying another (in essence also rank-revealing) orthogonal factorization to the nullspace matrix thus created, we can then generate triangular blocks with small norm in the lower right part ofR. This scheme is applied in an iterative fashion until the rank has been revealed in the (updated) QR factorization. We show that the algorithm produces the correct solution, under very weak assumptions for the orthogonal factorization used for the nullspace matrix. We then discuss issues concerning an efficient implementation of the algorithm and present some numerical experiments. Our experiments show that the block algorithm is reliable and successfully captures several small singular values, in particular in the initial block steps. Our experiments confirm the reliability of our algorithm and show that the block algorithm greatly reduces the number of triangular solves and increases the computational granularity of the RRQR computation.

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 9700 S. Cass Ave., 60439, Argonne, IL, USA

    Christian H. Bischof

  2. UNI.C (Danish Computing Center for Research and Education), Technical University of Denmark, Building 305, DK-2800, Lyngby, Denmark

    Per Christian Hansen

Authors
  1. Christian H. Bischof
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  2. Per Christian Hansen
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Additional information

Communicated by M.H. Gutknecht

This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38. The second author was also sponsored by a travel grant from the Knud Højgaards Fond.

This work was partially completed while the author was visiting the IBM Scientific Center in Heidelberg, Germany.

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Bischof, C.H., Hansen, P.C. A block algorithm for computing rank-revealing QR factorizations. Numer Algor 2, 371–391 (1992). https://doi.org/10.1007/BF02139475

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