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

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.

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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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Correspondence to Sven-Erik Ekström.

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Ahmad, F., Al-Aidarous, E., Alrehaili, D. et al. Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?. Numer Algor 78, 867–893 (2018). https://doi.org/10.1007/s11075-017-0404-z

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Keywords

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

Mathematics Subject Classifications (2010)

  • 15B05
  • 65F15
  • 65D05
  • 65B05