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Abstract
In this work, we investigate the spectra of "flipped" Toeplitz sequences, i.e., the asymptotic spectral behaviour of {Y_{n}T_{n}(f)}_{n}, where T_{n}(f)∈R^{n×n} is a real Toeplitz matrix generated by a function f∈L^{1}([π,π]), and Y_{n} is the exchange matrix, with 1s on the main antidiagonal. We show that the eigenvalues of Y_{n}T_{n}(f) are asymptotically described by a 2×2 matrixvalued function, whose eigenvalue functions are ±f. It turns out that roughly half of the eigenvalues of Y_{n}T_{n}(f) are well approximated by a uniform sampling of f over [π,π] 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.
Original language  English 

Number of pages  19 
Journal  BIT Numerical Mathematics 
Publication status  Accepted/In press  22 Nov 2018 
Keywords
 Toeplitz matrices
 spectral theory
 GLT theory
 Hankel matrices
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Dive into the research topics of 'Spectral properties of flipped Toeplitz matrices and related preconditioning'. Together they form a unique fingerprint.Projects
 1 Finished

Effective preconditioners for linear systems in fractional diffusion
EPSRC (Engineering and Physical Sciences Research Council)
19/01/18 → 19/06/20
Project: Research