The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

Abstract

Given a Lebesgue integrable function f over [0, 2π], we consider the sequence of matrices {Y n T n [f ]} n , where T n [f ] is the n-by-n Toeplitz matrix generated by f and Y n is the flip permutation matrix, also called the anti-identity matrix. Because of the unitary character of Y n , the singular values of T n [f ] and Y n T n [f ] coincide. However, the eigenvalues are affected substantially by the action of the matrix Y n . Under the assumption that the Fourier coefficients are real, we prove that {Y n T n [f ]} n is distributed in the eigenvalue sense aswith g(θ) = |f (θ)|. We also consider the preconditioning introduced by Pestana and Wathen and, by using the same arguments, we prove that the preconditioned sequence is distributed in the eigenvalue sense as φ 1 , under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.

Preconditioners for Krylov subspace methods: An overview

“…This well‐known fact has been exploited to develop solvers that are typically faster than GMRES applied to the original system, thanks to the short‐term recurrences of MINRES [364‐366]. We stress that Y does not, in general, improve the convergence rate [367,368]; a second, symmetric positive definite, preconditioner is almost always required. However, in contrast to the nonsymmetric system, the preconditioner choice can be guided by MINRES convergence theory, and mathematical guarantees of fast convergence can be obtained.…”

Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioning

“…Indeed, we also consider a class of preconditioners whose efficiency is motivated by the theoretical results in. 3 In all the examples, we consider a Hermitian positive definite circulant matrix. In its construction the concepts of absolute value circulant matrix and Frobenius optimal preconditioner are involved.…”

“…In the following, we report the spectral features of symmetrized Toeplitz sequences as they were presented in Reference 3. First, we give the following definition.…”

“…Note that, since Y n is a unitary matrix, the singular values of Y n T n [ f ] and T n [ f ] are identical, hence the singular value distributions of { Y n T n [ f ]} n and { T n [ f ]} n are the same. The results on the spectral distribution of { Y n T n [ f ]} n were obtained independently in References 3 and 4. In Reference 3, the authors prove that { Y n T n [ f ]} n is distributed in the eigenvalue sense as under the assumptions that f belongs to and has real Fourier coefficients.…”

“…In Reference 3, the authors prove that { Y n T n [ f ]} n is distributed in the eigenvalue sense as under the assumptions that f belongs to and has real Fourier coefficients. Furthermore, few preconditioning strategies and a study of the spectra of the related preconditioned matrix‐sequences are presented in References 3,4. We highlight that functions of Toeplitz matrices have crucial relevance in several applications.…”