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A note on the recursive calculation of dominant singular subspaces

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Abstract

A recursive procedure for computing an approximation of the left and right dominant singular subspaces of a given matrix is proposed in [1]. The method is particularly suited for matrices with many more rows than columns. The procedure consists of a few steps. In one of these steps a Householder transformation is multiplied to an upper triangular matrix. The following step consists in recomputing an upper triangular matrix from the latter product. In [1] it is said that the latter step is accomplished in O(k3) operations, where k is the order of the triangular matrix. In this short note we show that this step can be accomplished in O(k2) operations.

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References

  1. Y. Chahlaoui, K. Gallivan and P. Van Dooren, Recursive calculation of dominant singular subspaces, SIAM J. Matrix Anal. Appl. 25(2) (2004) 445–463.

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  2. T.F. Chan, An improved algorithm for computing the singular value decomposition, ACM Trans. Math. Software 8(1) (1982) 72–83.

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Authors and Affiliations

  1. Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, via G. Amendola 122/D, I–70126, Bari, Italy

    Nicola Mastronardi

  2. Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001, Leuven (Heverlee), Belgium

    Marc Van Barel & Raf Vandebril

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

Correspondence to Nicola Mastronardi.

Additional information

Communicated by P. Van Dooren

This research was partially supported by MIUR, grant number 2002014121 (first author) and 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) (second and third author). The scientific responsibility rests with the authors.

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15A15, 15A09, 15A23

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Mastronardi, N., Van Barel, M. & Vandebril, R. A note on the recursive calculation of dominant singular subspaces. Numer Algor 38, 237–242 (2005). https://doi.org/10.1007/s11075-004-4338-x

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  • DOI: https://doi.org/10.1007/s11075-004-4338-x

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