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

TL;DR

This paper analyzes the asymptotic spectral behavior of flipped multilevel Toeplitz matrices and certain preconditioned sequences, revealing that their spectra are governed by a 2x2 matrix function related to the generating function.

Contribution

It extends spectral analysis to flipped multilevel Toeplitz matrices and preconditioned sequences, providing explicit spectral descriptions and eigenvalue distributions.

Findings

Spectral behavior determined by a 2x2 matrix function with eigenvalues ±|f|

Characterization of eigenvalue distribution for preconditioned sequences

Numerical experiments validating theoretical results

Abstract

In this work, we perform a spectral analysis of flipped multilevel Toeplitz sequences, i.e., we study the asymptotic spectral behaviour of $\{Y_{\boldsymbol{n}}T_{\boldsymbol{n}}(f)\}_{\boldsymbol{n}}$, where $T_{\boldsymbol{n}}(f)$ is a real, square multilevel Toeplitz matrix generated by a function $f\in L^1([-\pi,\pi]^d)$ and $Y_{\boldsymbol{n}}$ is the exchange matrix, which has $1$s on the main anti-diagonal. In line with what we have shown for unilevel flipped Toeplitz matrix sequences, the asymptotic spectrum is determined by a $2\times 2$ 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.