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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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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DOI: https://doi.org/10.1007/s11075-017-0404-z
Keywords
- (Preconditioned) Toeplitz matrix
- Mass and stiffness matrix
- Eigenvalues
- Eigenvalue asymptotics
- Polynomial interpolation
- Extrapolation