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Numerical Algorithms

, Volume 78, Issue 3, pp 867–893 | Cite as

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

  • Fayyaz Ahmad
  • Eman Salem Al-Aidarous
  • Dina Abdullah Alrehaili
  • Sven-Erik EkströmEmail author
  • Isabella Furci
  • Stefano Serra-Capizzano
Open Access
Original Paper

Abstract

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {T n (f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {T n−1(g)T n (f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

Keywords

(Preconditioned) Toeplitz matrix Mass and stiffness matrix Eigenvalues Eigenvalue asymptotics Polynomial interpolation Extrapolation 

Mathematics Subject Classifications (2010)

15B05 65F15 65D05 65B05 

Notes

Acknowledgements

The research of Eman Salem Al-Aidarous was funded by King Abdulaziz University during scientific communication year 2017–2018. The research of Sven-Erik Ekström is cofinanced by the Graduate School in Mathematics and Computing (FMB) and Uppsala University. The research of the Isabella Furci and Stefano Serra-Capizzano is cofinanced by INdAM-GNCS (Istituto Nazionale di Alta Matematica - Gruppo Nazionale di Calcolo Scientifico).

Finally, a special thanks to the referee for pertinent comments, which helped us to improve the quality of the paper.

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Science and High TechnologyUniversity of InsubriaComoItaly
  2. 2.Faculty of Science, Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Division of Scientific Computing, Department of Information Technology, ITCUppsala UniversityUppsalaSweden

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