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

  • Carlo GaroniEmail author
  • Stefano Serra-Capizzano
  • Debora Sesana
Chapter
Part of the Springer INdAM Series book 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 An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the discretization parameter n tends to infinity, these matrices An give rise to a sequence {An}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.

Keywords

Singular values and eigenvalues Block generalized locally Toeplitz sequences Block Toeplitz matrices Discretization of differential equations 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Carlo Garoni
    • 1
    • 2
    Email author
  • Stefano Serra-Capizzano
    • 2
    • 3
  • Debora Sesana
    • 2
  1. 1.University of Italian SwitzerlandInstitute of Computational ScienceLuganoSwitzerland
  2. 2.University of InsubriaDepartment of Science and High TechnologyComoItaly
  3. 3.Uppsala UniversityDepartment of Information Technology, Division of Scientific ComputingUppsalaSweden

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