×

Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences. (English) Zbl 1465.15036

Summary: In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that \(f\) belongs to \(L^1([-\pi,\pi])\) and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence \(\{Y_nT_n[f]\}_n\) has been identified, where \(n\) is the matrix size, \(Y_n\) is the anti-identity matrix, and \(T_n[f]\) is the Toeplitz matrix generated by \(f\). In this note, the authors consider the multilevel Toeplitz matrix \(T_\mathbf{n}[f]\) generated by \(f\in L^1([-\pi,\pi]^k)\), \(\mathbf{n}\) being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrix-sequence \(\{Y_\mathbf{n}T_\mathbf{n}[f]\}_\mathbf{n}\) with \(Y_\mathbf{n}\) being the corresponding tensorization of the anti-identity matrix.

MSC:

15B05 Toeplitz, Cauchy, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] F. Avram. On bilinear forms in Gaussian random variables and Toeplitz matrices.Prob. Theory Related Fields, 79(1):37-45, 1988. · Zbl 0648.60043
[2] R. Bhatia.Matrix Analysis. Springer, New York, 1997. · Zbl 0863.15001
[3] R. Chan and X. Jin.An Introduction to Iterative Toeplitz Solvers,Vol. 5. Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. · Zbl 1146.65028
[4] D. Fasino and P. Tilli. Spectral clustering properties of block multilevel Hankel matrices.Linear Algebra Appl., 306(1- 3):155-163, 2000. · Zbl 0952.15016
[5] P. Ferrari, I. Furci, S. Hon, M. Mursaleen, and S. Serra-Capizzano. The eigenvalue distribution of special 2-by-2 block matrix-sequences with applications to the case of symmetrized Toeplitz structures.SIAM J. Matrix Anal. Appl., 40(3):1066-1086, 2019. · Zbl 1426.15041
[6] P. Ferrari, N. Barakitis, and S. Serra-Capizzano. Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioningNumer. Linear Algebra Appl., 28:e2332, 2020. · Zbl 07332746
[7] C. Garoni and S. Serra-Capizzano.Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I. Springer, Cham, 2017. · Zbl 1376.15002
[8] C. Garoni and S. Serra-Capizzano.Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II. Springer, Cham, 2018. · Zbl 1448.47004
[9] C. Garoni, S. Serra-Capizzano, and P. Vassalos. A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences.Oper. Matrices, 9(3):549-561, 2015. · Zbl 1329.47019
[10] U. Grenander and G. Szeg˝o.Toeplitz Forms and Their Applications, second edition. Chelsea Publishing Co., New York, 1984. · Zbl 0611.47018
[11] W. HackbuschIterative Solution of Large Sparse Systems of Equations. Springer-Verlag, New York, 2016. · Zbl 1347.65063
[12] M. Mazza and J. Pestana. Spectral properties of flipped Toeplitz matrices.BIT Numer. Math., 59:463-482, 2019. · Zbl 1417.15045
[13] M. Mazza and J. Pestana. The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings. arXiv:2011.08372, 2020.
[14] M. Ng.Iterative Methods for Toeplitz Systems. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2004. · Zbl 1059.65031
[15] S. Parter. On the distribution of the singular values of Toeplitz matrices.Linear Algebra Appl., 80:115-130, 1986. · Zbl 0601.15006
[16] J. Pestana. Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices.SIAM J. Matrix Anal. Appl., 40(3):870-887, 2019. · Zbl 1420.65024
[17] J. Pestana and A. Wathen. A preconditioned MINRES method for nonsymmetric Toeplitz matrices.SIAM J. Matrix Anal. Appl., 36(1):273-288, 2015. · Zbl 1315.65034
[18] Y. SaadIterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. · Zbl 1031.65046
[19] S. Serra-Capizzano. Distribution results on the algebra generated by Toeplitz sequences: a finite-dimensional approach. Linear Algebra Appl., 328(1):121-130, 2001. · Zbl 1003.15008
[20] S. Serra-Capizzano.Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations.Linear Algebra Appl., 366:371-402, 2003. · Zbl 1028.65109
[21] S. Serra-Capizzano. The GLT class as a generalized Fourier analysis and applications.Linear Algebra Appl., 419(1):180- 233, 2006. · Zbl 1109.65032
[22] S. Serra-Capizzano and P. Tilli. On unitarily invariant norms of matrix-valued linear positive operators.J. Inequalities Appl., 7(3):309-330, 2002. · Zbl 1055.15039
[23] P. Tilli. A note on the spectral distribution of Toeplitz matrices.Linear and Multilinear Algebra, 45(2-3):147-159, 1998. · Zbl 0951.65033
[24] P. Tilli. Some results on complex Toeplitz eigenvalues.J. Math. Anal. Appl., 239(2):390-401, 1999. · Zbl 0935.15002
[25] E. Tyrtyshnikov. New theorems on the distribution of eigenvalues and singular values of multilevel Toeplitz matrices. Doklady Akademii Nauk, 333(3):300-303, 1993.
[26] E. Tyrtyshnikov. Influence of matrix operations on the distribution of eigenvalues and singular values of Toeplitz matrices. Linear Algebra Appl., 207:225-249, 1994. · Zbl 0813.15005
[27] E. Tyrtyshnikov. A unifying approach to some old and new theorems on distribution and clustering.Linear Algebra Appl., 232:1-43, 1996. · Zbl 0841.15006
[28] N. Zamarashkin and E. Tyrtyshnikov. Distribution of the eigenvalues and singular numbers of Toeplitz matrices under weakened requirements on the generating function.Matematicheskii Sbornik, 188(8):83-92, 1997 · Zbl 0898.15007
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.