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Rational QR-iteration without inversion

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Abstract

In this manuscript a new method will be presented for performing a QR-iteration with (A − σ I)(A − κ I)−1 = QR without explicit inversion of the factor (A − κ I)−1. A QR-method driven by a rational function is attractive since convergence can occur at both sides of the matrix. Each step of this new iteration consists of two substeps. In the explicit version, first an RQ-factorization of the initial matrix A − κ I = RQ will be computed, followed by a QR-factorization of the matrix (A − σ I)Q H. The factorization of (A − σ I)Q H can be computed in an intelligent manner, exploiting properties of the already known RQ-factorization of A − κ I. The similarity transformation yielding the QR-step is defined by the unitary factor Q in the QR-factorization of the transformed matrix (A − σ I)Q H. Examples will be given, illustrating how to efficiently compute the factorization for some specific classes of matrices. The novelties of this approach with respect to these matrix classes will be discussed.

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Authors and Affiliations

  1. Department of Computer Science, KULeuven, Heverlee, 3001, Leuven, 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

The research of the authors R. Vandebril and M. Van Barel, was partially supported by the Research Council K.U.Leuven, project OT/05/40 (Large rank structured matrix computations), CoE EF/05/006 Optimization in Engineering (OPTEC), by the Fund for Scientific Research–Flanders (Belgium), G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), G.0423.05 (RAM: Rational modelling: optimal conditioning and stable algorithms), and by the Interuniversity Poles of Attraction Poles, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). R. Vandebril has a grant as “Postdoctoraal Onderzoeker” from the Fund for Scientific Research–Flanders (Belgium). The work of N. Mastronardi was partially supported by MIUR, grant number 2004015437. The scientific responsibility rests with the authors.

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Vandebril, R., Van Barel, M. & Mastronardi, N. Rational QR-iteration without inversion. Numer. Math. 110, 561–575 (2008). https://doi.org/10.1007/s00211-008-0177-3

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  • DOI: https://doi.org/10.1007/s00211-008-0177-3

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