Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioning. Issue 1 (15th September 2020)

• Record Type: Journal Article
• Authors: Ferrari, Paola; Barakitis, Nikos; Serra‐Capizzano, Stefano
• Is Part Of: Numerical linear algebra with applications. Volume 28:Issue 1(2021)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2020-09-15
• Subjects: eigenvalue distribution -- functions of matrices -- preconditioning -- singular value distribution -- Toeplitz matrices; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2332
• Languages: English
• ISSNs: 1070-5325

Abstract

Abstract: The singular value distribution of the matrix‐sequence { Y n T n [ f ]} n, with T n [ f ] generated by f ∈ L 1 ( [ − 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 { Y n T n [ 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 { Y n T n [ f ]} n is distributed in the eigenvalue sense as ϕ | f | ( θ ) = | f ( θ ) |, θ ∈ [ 0, 2 π ], − | f ( − θ ) |, θ ∈ [ − 2 π, 0 ), under the assumptions that f belongs to L 1 ( [ − 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 ( T n [ f ])} n, where h is an analytic function. In particular, we provide the singular value distribution of the sequence { h ( T n [ f ])} n, the eigenvalue distribution of the sequence { Y n h ( T n [ 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.