Abstract
In this work, we investigate the spectra of “flipped” Toeplitz sequences, i.e., the asymptotic spectral behaviour of {Y_nT_n(f)}_n, where T_n(f)in mathbb {R}^{ntimes n} is a real Toeplitz matrix generated by a function fin L^1([-pi ,pi ]), and Y_n is the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues of Y_nT_n(f) are asymptotically described by a 2times 2 matrix-valued function, whose eigenvalue functions are pm , |f|. It turns out that roughly half of the eigenvalues of Y_nT_n(f) are well approximated by a uniform sampling of |f| over [-,pi ,pi ], 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.
Highlights
Given a Toeplitz matrix Tn( f ) ∈ Rn×n generated by a function f ∈ L1([−π, π ]), and the exchange matrix Yn ∈ Rn×n
Since Yn Tn( f ) is symmetric, the resulting linear system may be solved by the MINRES method [18,21] or by preconditioned MINRES [16,17], with its descriptive convergence rate bounds based on eigenvalues
We leverage the block Generalized Locally Toeplitz (GLT) algebra as a black-box tool and we naturally get a matrix-valued symbol with two eigenvalue functions that, in line with Remark 2.1, immediately fits with the quasi-half positive/negative nature of the spectrum of Yn Tn( f )
Summary
Given a Toeplitz matrix Tn( f ) ∈ Rn×n generated by a function f ∈ L1([−π, π ]), and the exchange matrix Yn ∈ Rn×n,. In this paper we seek to explain this observation. One reason for characterizing the spectra of these flipped matrices relates to the solution of linear systems with Toeplitz coefficient matrices. Whilst there has been significant interest in relating the eigenvalues and singular values of Toeplitz sequences to generating functions, analogous results have not been proved for flipped Toeplitz sequences and corresponding preconditioned ones. This paper aims to fill this gap. 2 we describe the tools we require, we introduce the class of Generalized locally Toeplitz matrix-sequences and related properties [6]. 3. The main results, that describe the spectra of sequences of (preconditioned) flipped Toeplitz matrices can be found in Sect.
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