Spectral properties of flipped Toeplitz matrices and related preconditioning

Mazza, M. and Pestana, J. (2018) Spectral properties of flipped Toeplitz matrices and related preconditioning. BIT Numerical Mathematics. ISSN 0006-3835 (In Press)

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Abstract

In this work, we investigate the spectra of "flipped" Toeplitz sequences, i.e., the asymptotic spectral behaviour of {YnTn(f)}n, where Tn(f)∈Rn×n is a real Toeplitz matrix generated by a function f∈L1([-π,π]), and Yn is the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues of YnTn(f) are asymptotically described by a 2×2 matrix-valued function, whose eigenvalue functions are ±|f|. It turns out that roughly half of the eigenvalues of YnTn(f) are well approximated by a uniform sampling of |f| over [-π,π]  while the remaining are well approximated by a uniform sampling of -|f| over the same interval. When f vanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative. Some insights on the spectral distribution of related preconditioned sequences are provided as well. Finally, a wide number of numerical results illustrate our theoretical findings.

ORCID iDs

Mazza, M. and Pestana, J. ORCID logoORCID: https://orcid.org/0000-0003-1527-3178;
  • Item type:Article
    ID code:66212
    Dates:
    Date
    Event
    22 November 2018
    Published
    22 November 2018
    Accepted
    3 September 2018
    Submitted
    Keywords:Toeplitz matrices, spectral theory, GLT theory, Hankel matrices, Mathematics, Mathematics(all)
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:26 Nov 2018 11:50
    Last modified:18 Jan 2023 10:38
    Related URLs:
    URI:https://strathprints.strath.ac.uk/id/eprint/66212
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