×
Ekström, Sven-Erik; Garoni, Carlo; Serra-Capizzano, Stefano

Are the eigenvalues of banded symmetric Toeplitz matrices known in almost closed form? (English) Zbl 1405.15037

Exp. Math. 27, No. 4, 478-487 (2018).
Summary: J. M. Bogoya et al. [J. Math. Anal. Appl. 422, No. 2, 1308–1334 (2015; Zbl 1302.65086)] have recently obtained for the eigenvalues of a Toeplitz matrix, under suitable assumptions on the generating function, the precise asymptotic expansion as the matrix size goes to infinity. In this article we provide numerical evidence that some of these assumptions can be relaxed. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric Toeplitz matrices with a high level of accuracy and a relatively low computational cost.

Cited in 23 Documents

MSC:

15B05 Toeplitz, Cauchy, and related matrices
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65D05 Numerical interpolation
65B05 Extrapolation to the limit, deferred corrections

Keywords:

eigenvalue asymptotics; eigenvalues; extrapolation; polynomial interpolation; Toeplitz matrix

Citations:

Zbl 1302.65086
Cite Review PDF
Full Text: DOI Link

References:

[1] Barrera, [Barrera And Grudsky 17] M.; Grudsky, S. M., Asymptotics of Eigenvalues for Pentadiagonal Symmetric Toeplitz Matrices., Oper. Theory Adv. Appl., 259, 51-77 (2017) · Zbl 1365.15040
[2] Bauer, [Bauer 61] F. L., La méthode d’intégration numérique de Romberg · Zbl 0107.33706
[3] Bauer, [Bauer Et Al. 63] F. L.; Rutishauser, H.; Stiefel, E., New Aspects in Numerical Quadrature., Proc. Symp. Appl. Math., 15, 199-218 (1963) · Zbl 0133.09201
[4] Bevilacqua, [Bevilacqua Et Al. 92] R.; Bini, D.; Capovani, M.; Menchi, O., Metodi Numerici (1992), Bologna, Italy: Zanichelli, Bologna, Italy
[5] Bhatia, [Bhatia 97] R., Matrix Analysis (1997), New York: Springer, New York
[6] Bini, [Bini And Capovani 83] D.; Capovani, M., Spectral and Computational Properties of Band Symmetric Toeplitz Matrices., Linear Algebra Appl., 52-53, 99-126 (1983) · Zbl 0549.15005
[7] Bogoya, [Bogoya Et Al. 15A] J. M.; Böttcher, A.; Grudsky, S. M.; Maximenko, E. A., Eigenvalues of Hermitian Toeplitz Matrices with Smooth Simple-loop Symbols, J. Math. Anal. Appl., 422, 1308-1334 (2015) · Zbl 1302.65086
[8] Bogoya, [Bogoya Et Al. 15B] J. M.; Böttcher, A.; Grudsky, S. M.; Maximenko, E. A., Maximum Norm Versions of the Szegő and Avram-Parter Theorems for Toeplitz Matrices., J. Approx. Theory, 196, 79-100 (2015) · Zbl 1320.15028
[9] Bogoya, [Bogoya Et Al. 16] J. M.; Böttcher, A.; Maximenko, E. A., From Convergence in Distribution to Uniform Convergence., Bol. Soc. Mat. Mex., 22, 695-710 (2016) · Zbl 1419.60005
[10] Bogoya, [Bogoya Et Al. 17] J. M.; Grudsky, S. M.; Maximenko, E. A., Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness., Oper. Theory Adv. Appl., 259, 179-212 (2017) · Zbl 1468.15007
[11] Böttcher, [Böttcher Et Al. 10] A.; Grudsky, S. M.; Maximenko, E. A., Inside the Eigenvalues of Certain Hermitian Toeplitz Band Matrices., J. Comput. Appl. Math., 233, 2245-2264 (2010) · Zbl 1195.15009
[12] Böttcher, [Böttcher And Silbermann 99] A.; Silbermann, B., Introduction to Large Truncated Toeplitz Matrices (1999), New York: Springer, New York · Zbl 0916.15012
[13] Brezinski, [Brezinski And Redivo Zaglia 91] C.; Zaglia, M. Redivo, Extrapolation Methods: Theory and Practice (1991), North-Holland: Elsevier Science Publishers B.V., North-Holland · Zbl 0744.65004
[14] Davis, [Davis 75] P. J., Interpolation and Approximation (1975), Mineola, NY: Dover Publications, Mineola, NY · Zbl 0329.41010
[15] Ekström, [Ekström And Serra-Capizzano] S.-E.; Serra-Capizzano, S., Eigenvalues and Eigenvectors of Banded Toeplitz Matrices and the Related Symbols · Zbl 1513.65095
[16] Garoni, [Garoni And Serra-Capizzano 17] C.; Serra-Capizzano, S., Generalized Locally Toeplitz Sequences: Theory and Applications (Volume I) (2017), Cham, Switzerland: Springer International Publishing AG, Cham, Switzerland · Zbl 1376.15002
[17] Serra-Capizzano, [Serra-Capizzano 96] S., On the Extreme Spectral Properties of Toeplitz Matrices Generated by \(\textit{L^1\) · Zbl 0851.15008
[18] Stoer, [Stoer And Bulirsch 02] J.; Bulirsch, R., Introduction to Numerical Analysis (2002), New York: Springer, New York · Zbl 1004.65001
[19] Tilli, [Tilli 98] P., A Note on the Spectral Distribution of Toeplitz Matrices., Linear Multilinear Algebra, 45, 147-159 (1998) · Zbl 0951.65033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.