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A QR-type reduction for computing the SVD of a general matrix product/quotient

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Abstract

In this paper, a QR-type reduction technique is developed for the computation of the SVD of a general matrix product/quotient \(A=A_1^{s_1}A_2^{s_2}\cdots A_m^{s_m}\) with \(A_i\in {\bf R}^{n \times n} \hbox{ and } s_i=1\)or \(s_i=-1\). First the matrix A is reduced by at most m QR-factorizations to the form \(Q^{(1)}_{11}(Q^{(1)}_{21})^{-1}\), where \(Q^{(1)}_{11}, Q^{(1)}_{21}\in {\bf R}^{n \times n}\) and \((Q^{(1)}_{11})^TQ^{(1)}_{11}+(Q^{(1)}_{21})^TQ^{(1)}_{21}=I\). Then the SVD of A is obtained by computing the CSD (Cosine-Sine Decomposition) of \(Q^{(1)}_{11}\) and \(Q^{(1)}_{21}\) using the Matlab command gsvd. The performance of the proposed method is verified by some numerical examples.

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543

    Delin Chu

  2. ETIS, UPRES-A 8051, ENSEA / UCP, 6, avenue du Ponceau, BP 44, F 95014, Cergy-Pontoise Cedex, France

    Lieven De Lathauwer

  3. SCD-SISTA, Department of Electrical Engineering (ESAT), Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, B-3001, Leuven-Heverlee, Belgium

    Bart De Moor

Authors
  1. Delin Chu
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  2. Lieven De Lathauwer
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  3. Bart De Moor
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Correspondence to Delin Chu.

Additional information

Mathematics Subject Classification (1991): 65F15, 65H15

This work is supported by grants from several funding agencies: 1. Research Council KUL: Concerted Research Action GOA-Mefisto 666 (Mathematical Engineering) 2. Flemish Government: FWO (Fund for Scientific Research - Flanders) projects G292.95 and G256.97, FWO Research Communities ICCoS and ANMMM, 3. Belgian Federal Government: DWTC (IUAP IV-02 and IUAP V-10-29).

Part of this research was carried out while D. Chu was a visiting researcher at the K.U.Leuven. L. De Lathauwer holds a permanent research position with the French CNRS; he also holds a honorary post-doctoral research mandate with the FWO. B. De Moor is a full professor at the K.U.Leuven. The scientific responsibility is assumed by the authors

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Chu, D., Lathauwer, L. & Moor, B. A QR-type reduction for computing the SVD of a general matrix product/quotient. Numer. Math. 95, 101–121 (2003). https://doi.org/10.1007/s00211-002-0431-z

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  • DOI: https://doi.org/10.1007/s00211-002-0431-z

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