Theoretical results for eigenvalues, singular values, and eigenvectors of (flipped) Toeplitz matrices and related computational proposals

TL;DR

This paper provides new theoretical insights into the eigenvalues, singular values, and eigenvectors of (flipped) Toeplitz matrices, addressing both asymptotic and fixed-size cases, with implications for efficient spectral computation and linear system solutions.

Contribution

It introduces novel theoretical results for the spectral properties of (flipped) Toeplitz matrices, including non-monotone generating functions, and proposes matrix-less algorithms for spectral computation.

Findings

Spectral behavior characterized for large and fixed matrices.

Effective matrix-less algorithms for spectrum computation.

Numerical experiments validate theoretical results.

Abstract

In a series of recent papers the spectral behavior of the matrix sequence $\{Y_nT_n(f)\}$ is studied in the sense of the spectral distribution, where $Y_n$ is the main antidiagonal (or flip matrix) and $T_n(f)$ is the Toeplitz matrix generated by the function $f$, with $f$ being Lebesgue integrable and with real Fourier coefficients. This kind of study is also motivated by computational purposes for the solution of the related large linear systems using the (preconditioned) MINRES algorithm. Here we complement the spectral study with more results holding both asymptotically and for a fixed dimension $n$, and with regard to eigenvalues, singular values, and eigenvectors of $T_n(f), Y_nT_n(f)$ and to several relationships among them: beside fast linear solvers, a further target is the design of ad hoc procedures for the computation of the related spectra via matrix-less algorithms, with a cost being linear in the number of computed eigenvalues. We emphasize that the challenge of the case of non-monotone generating functions is considered in the current work, for which the previous matrix-less algorithms fail. Numerical experiments are reported and commented, with the aim of showing in a visual way the theoretical analysis.