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Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioning
Author(s) -
Ferrari Paola,
Barakitis Nikos,
SerraCapizzano Stefano
Publication year - 2021
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2332
Subject(s) - toeplitz matrix , mathematics , symmetrization , eigenvalues and eigenvectors , singular value , combinatorics , sequence (biology) , matrix (chemical analysis) , distribution (mathematics) , function (biology) , mathematical analysis , pure mathematics , physics , quantum mechanics , materials science , evolutionary biology , biology , composite material , genetics
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 asϕ | f | ( θ ) =| f ( θ ) | , θ ∈ [ 0 , 2 π ] ,− | f ( − θ ) | , θ ∈ [ − 2 π , 0 ) ,under the assumptions that f belongs toL 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.