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A QR–method for computing the singular values via semiseparable matrices

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Summary.

The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a new method for computing the singular value decomposition of a real matrix. In a first phase, an algorithm for reducing the matrix A into an upper triangular semiseparable matrix by means of orthogonal transformations is described. A remarkable feature of this phase is that, depending on the distribution of the singular values, after few steps of the reduction, the largest singular values are already computed with a precision depending on the gaps between the singular values. An implicit QR–method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values. The numerical tests show that the proposed method can compete with the standard method (using an intermediate bidiagonal matrix) for computing the singular values of a matrix.

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Authors and Affiliations

  1. Department of Computer Science,  , 3001, Leuven (Heverlee), Belgium

    Raf Vandebril & Marc Van Barel

  2. Istituto per le Applicazioni del Calcolo M. Picone, sez. Bari, Italy

    Nicola Mastronardi

Authors
  1. Raf Vandebril
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  2. Marc Van Barel
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  3. Nicola Mastronardi
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Corresponding author

Correspondence to Marc Van Barel.

Additional information

Mathematics Subject Classification (2000): 65F15, 15A18

The research of the first two authors was partially supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The work of the third author was partially supported by MIUR, grant number 2002014121. The scientific responsibility rests with the authors.

Acknowledgments.We thank the referees for their suggestions which increased the readability of the paper.

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Vandebril, R., Barel, M. & Mastronardi, N. A QR–method for computing the singular values via semiseparable matrices. Numer. Math. 99, 163–195 (2004). https://doi.org/10.1007/s00211-004-0550-9

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  • DOI: https://doi.org/10.1007/s00211-004-0550-9

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