Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory: The preconditioned setting

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form  ...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 466; p. 128483
Main Authors: Bogoya, M., Serra-Capizzano, S., Vassalos, P.
Format: Journal Article
Language:English
Published: 01.04.2024
Subjects:
Asymptotic expansion
Beräkningsvetenskap med inriktning mot numerisk analys
Numerical algorithm
Preconditioned matrix
Scientific Computing with specialization in Numerical Analysis
Spectra
Toeplitz matrix
ISSN:0096-3003, 1873-5649
Online Access:Get full text
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Summary:Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form  with  real-valued, g nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that  is even and monotonic over , matrix-less algorithms have been developed for the fast eigenvalue computation of large preconditioned matrices of the type above, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions as in the case , combined with the extrapolation idea, and hence we conjecture that the simple-loop theory has to be extended in such a new setting, as the numerics strongly suggest. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we consider new matrix-less algorithms ad hoc for the current case. Numerical experiments show a much higher accuracy till machine precision and the same linear computational cost, when compared with the matrix-less procedures already proposed in the literature.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2023.128483