The singular value distribution of the matrix-sequence {YnTn[f]}n, with Tn[f] generated by (Formula presented.), was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273-288]. The results on the spectral distribution of {YnTn[f]}n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463-482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S. Serra-Capizzano, SIAM J. Matrix Anal. Appl., 40(3):1066-1086, 2019]. In the latter reference, the authors prove that {YnTn[f]}n is distributed in the eigenvalue sense as (Formula presented.) under the assumptions that f belongs to (Formula presented.) and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix-sequences of the form {h(Tn[f])}n, where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(Tn[f])}n, the eigenvalue distribution of the sequence {Ynh(Tn[f])}n, and the conditions on f and h for these distributions to hold. Finally, the implications of our findings are discussed, in terms of preconditioning and of fast solution methods for the related linear systems.

Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioning

Serra Capizzano S.
2021-01-01

Abstract

The singular value distribution of the matrix-sequence {YnTn[f]}n, with Tn[f] generated by (Formula presented.), was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273-288]. The results on the spectral distribution of {YnTn[f]}n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463-482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S. Serra-Capizzano, SIAM J. Matrix Anal. Appl., 40(3):1066-1086, 2019]. In the latter reference, the authors prove that {YnTn[f]}n is distributed in the eigenvalue sense as (Formula presented.) under the assumptions that f belongs to (Formula presented.) and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix-sequences of the form {h(Tn[f])}n, where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(Tn[f])}n, the eigenvalue distribution of the sequence {Ynh(Tn[f])}n, and the conditions on f and h for these distributions to hold. Finally, the implications of our findings are discussed, in terms of preconditioning and of fast solution methods for the related linear systems.
2021
eigenvalue distribution; functions of matrices; preconditioning; singular value distribution; Toeplitz matrices
Ferrari, P.; Barakitis, N.; Serra Capizzano, S.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2119499
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