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

• Published in: Numerical linear algebra with applications 2021-01, Vol.28 (1), p.n/a
• Main Authors: Ferrari, Paola, Barakitis, Nikos, Serra‐Capizzano, Stefano
• Format: Article
• Language: English
• Subjects: Analytic functions; eigenvalue distribution; Eigenvalues; functions of matrices; Linear systems; Mathematical analysis; Matrix methods; Preconditioning; singular value distribution; Toeplitz matrices
• ISSN: 1070-5325; 1099-1506; 1099-1506
• DOI: 10.1002/nla.2332

The singular value distribution of the matrix‐sequence {YnTn[f]}n, with Tn[f] generated by f∈L1([−π,π]), 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 ϕ|f|(θ)=|f(θ)|,θ∈[0,2π],−|f(−θ)|,θ∈[−2π,0),under the assumptions that f belongs to L1([−π,π]) 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.