Given a Lebesgue integrable function f over [−π, π], we consider the sequence of matrices {YnTn[f]}n, where Tn[f] is the n-by-n Toeplitz matrix generated by f and Yn is the anti-identity matrix. Because of the unitary nature of Yn, the singular values of Tn[f] and YnTn[f] coincide. However, the eigenvalues are affected substantially by the action of Yn. Under the assumption that the Fourier coefficients of f are real, we prove that {YnTn[f]}n is distributed in the eigenvalue sense as ±|f|. A generalization of this result to the block Toeplitz case is also shown. We also consider the preconditioning introduced by [J. Pestana and A. Wathen, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 273–288] and prove that the preconditioned matrix-sequence is distributed in the eigenvalue sense as φ1 under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.

The eigenvalue distribution of special 2-by-2 block matrix-sequences with applications to the case of symmetrized toeplitz structures∗

Ferrari P.;Furci I.
;
Serra-Capizzano S.
2019-01-01

Abstract

Given a Lebesgue integrable function f over [−π, π], we consider the sequence of matrices {YnTn[f]}n, where Tn[f] is the n-by-n Toeplitz matrix generated by f and Yn is the anti-identity matrix. Because of the unitary nature of Yn, the singular values of Tn[f] and YnTn[f] coincide. However, the eigenvalues are affected substantially by the action of Yn. Under the assumption that the Fourier coefficients of f are real, we prove that {YnTn[f]}n is distributed in the eigenvalue sense as ±|f|. A generalization of this result to the block Toeplitz case is also shown. We also consider the preconditioning introduced by [J. Pestana and A. Wathen, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 273–288] and prove that the preconditioned matrix-sequence is distributed in the eigenvalue sense as φ1 under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.
2019
Circulant preconditioners; Eigenvalue distribution; Hankel matrices; Singular value distribution; Toeplitz matrices
Ferrari, P.; Furci, I.; Hon, S.; Mursaleen, M. A.; Serra-Capizzano, S.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2114407
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