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Structured Matrices in Numerical Linear Algebra pp 59–79Cite as

Block Generalized Locally Toeplitz Sequences: Topological Construction, Spectral Distribution Results, and Star-Algebra Structure

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 30))

Abstract

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic singular value and eigenvalue distribution of matrices A n arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the discretization parameter n tends to infinity, these matrices A n give rise to a sequence {A n}n, which often turns out to be a GLT sequence or one of its ‘relatives’, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences is still missing. The purpose of the present paper is to develop this theory in a systematic way.

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Acknowledgements

Carlo Garoni is a Marie-Curie fellow of the Italian INdAM under grant agreement PCOFUND-GA-2012-600198. The work of the authors has been supported by the INdAM GNCS (Gruppo Nazionale per il Calcolo Scientifico). The authors wish to thank Giovanni Barbarino for useful discussions.

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Authors and Affiliations

  1. University of Italian Switzerland, Institute of Computational Science, Lugano, Switzerland

    Carlo Garoni

  2. University of Insubria, Department of Science and High Technology, Como, Italy

    Carlo Garoni, Stefano Serra-Capizzano & Debora Sesana

  3. Uppsala University, Department of Information Technology, Division of Scientific Computing, Uppsala, Sweden

    Stefano Serra-Capizzano

Authors
  1. Carlo Garoni
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  2. Stefano Serra-Capizzano
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  3. Debora Sesana
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Corresponding author

Correspondence to Carlo Garoni .

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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Dario Andrea Bini

  2. Department of Mathematics, University of Genoa, Genoa, Italy

    Fabio Di Benedetto

  3. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

    Eugene Tyrtyshnikov

  4. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Marc Van Barel

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Garoni, C., Serra-Capizzano, S., Sesana, D. (2019). Block Generalized Locally Toeplitz Sequences: Topological Construction, Spectral Distribution Results, and Star-Algebra Structure. In: Bini, D., Di Benedetto, F., Tyrtyshnikov, E., Van Barel, M. (eds) Structured Matrices in Numerical Linear Algebra. Springer INdAM Series, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-030-04088-8_3

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