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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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References

  1. E. Anderson, Z. Bai, C. Bischof, J. W. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen: LAPACK Users' Guide, Society for Industrial and Applied Mathematics, Philadelphia, (1992)

  2. Z. Bai, J. W. Demmel: Computing the generalized singular value decomposition, SIAM J. Sci. Comput., 14, 1464–1486 (1993)

    Google Scholar 

  3. Z. Bai: The CSD, GSVD, their applications and computations, IMA Preprint 958, University of Minnesota, April 1992, Submitted to SIAM Rev

  4. D. Chu, L. De Lathauwer, B. De Moor: On the computation of Restricted Singular Value Decomposition via Cosine-Sine Decomposition. SIAM J. Matrix Anal. Appl., 22, 580–601 (2001)

    Google Scholar 

  5. B. De Moor, G. H. Golub: The restricted singular value decomposition: properties and applications. SIAM J. Matrix Anal. Appl., 12, 401–425 (1991)

    Google Scholar 

  6. B. De Moor: On the structure of generalized singular value and QR decompositions, SIAM J. Matrix Anal. Appl., 15, 347–358 (1994)

    Google Scholar 

  7. B. De Moor, P. Van Dooren: Generalizations of the QR and singular value decomposition, SIAM J. Matrix Anal. Appl., 13, 993–1014 (1992)

    Google Scholar 

  8. B. De Moor, G. H. Golub: Generalized singular value decompositions: a proposal for a standardized nomenclature, Internal Report 89-10, ESAT-SISTA, K.U.Leuven, Leuven, Belgium, (1989)

  9. K. Fernando, S. Hammarling: A product induced singular value decomposition for two matrices and balanced realisation, NAG Technical Report TR8/87.

  10. G. H. Golub, C. F. Van Loan: Matrix Computations, 3rd ed., The John Hopkins University Press, Baltimore, MD, (1996)

  11. G. Golub, K. Solna, P. Van Dooren: Computing the SVD of a general matrix product/quotient, SIAM J. Matrix Anal. Appl., to appear.

  12. L. Arnold, H. Crauel, J. Eckmann: Lyapunov Exponents, Spring-Verlag, Berlin, New York, (1991)

  13. M. T. Heath, A. J. Laub, C. C. Paige, R. C. Ward: Computing the singular value decomposition of a product of two matrices, SIAM J. Sci. Statist. Comput., 7, 1147–1159 (1986)

    Google Scholar 

  14. C. C. Paige, M. A. Saunders: Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18, 398–405 (1981)

    Google Scholar 

  15. C. C. Paige: Computing the generalized singular value decomposition, SIAM J. Sci. Stat. Comput., 7, 1126–1146 (1986)

    Google Scholar 

  16. G. W. Stewart: Computing the CS-decomposition of a partitioned orthogonal matrix, Numer. Math., 40, 297–306 (1982)

    Google Scholar 

  17. C. F. Van Loan: Generalizing the singular value decomposition, SIAM J. Numer. Anal., 13, 76–83 (1976)

    Google Scholar 

  18. C. F. Van Loan: Computing the CS and the generalized singular value decomposition, Numer. Math., 46, 479–491 (1985)

    Google Scholar 

  19. H. Zha: A numerical algorithm for computing the restricted singular value decomposition of matrix triplets, Linear Algebra Appl., 168, 1–25 (1992)

    Google Scholar 

  20. H. Zha: The restricted singular value decomposition of matrix triplets. SIAM J. Matrix Anal. Appl., 12, 172–194 (1991)

    Google Scholar 

  21. H. Zha: Computing the generalized singular values/vectors of large sparse or structured matrix pairs, Numerische Mathematik, 72, 391–417 (1996)

    Google Scholar 

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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.

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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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