The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings

M. Mazza, J. Pestana

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Abstract

In this work, we perform a spectral analysis of flipped multilevel Toeplitz sequences, i.e., we study the asymptotic spectral behavior of $\{Y_{{n}} T_{{n}} (f)\}_{{n}}$, where $T_{{n}}(f)$ is a real, square multilevel Toeplitz matrix generated by a function $f\in L^1([-\pi,\pi]^d)$ and $Y_n$ is the exchange matrix, which has 1's on the main antidiagonal. In line with what we have shown for unilevel flipped Toeplitz matrix sequences, the asymptotic spectrum is determined by a 2 x 2 matrix-valued function whose eigenvalues are $\pm |f|$. Furthermore, we characterize the eigenvalue distribution of certain preconditioned flipped multilevel Toeplitz sequences with an analysis that covers both multilevel Toeplitz and circulant preconditioners. Finally, all our findings are illustrated by several numerical experiments.
Original languageEnglish
Pages (from-to)1319-1336
Number of pages18
JournalSIAM Journal on Matrix Analysis and Applications
Volume42
Issue number3
DOIs
Publication statusPublished - 26 Aug 2021

Keywords

  • multilevel Toeplitz matrices
  • spectral symbol
  • GLT theory
  • preconditioning

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