2022
DOI: 10.48550/arxiv.2201.02024
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Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory

Abstract: Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f . Independently and under the milder hypothesis that f is even and monotonic over [0, π], matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by… Show more

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Cited by 1 publication
(2 citation statements)
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“…The numerical results are extremely precise, even compared with the already good performances described in [1,17,18,19,20] and of the same order as (or even better than) in [9] for the non preconditioned setting: in fact, it is not difficult to reach machine precision and the complexity for computing all the eigenvalues is still linear.…”
Section: Wherementioning
confidence: 69%
See 1 more Smart Citation
“…The numerical results are extremely precise, even compared with the already good performances described in [1,17,18,19,20] and of the same order as (or even better than) in [9] for the non preconditioned setting: in fact, it is not difficult to reach machine precision and the complexity for computing all the eigenvalues is still linear.…”
Section: Wherementioning
confidence: 69%
“…In the work [9] we used an asymptotic eigenvalue expansion which was based on the simple-loop theory (see for example [6,8,10] or the nice review [12]). However, the preconditioned setting has the additional complication of not having a formal supporting result.…”
Section: Algorithmic Proposalsmentioning
confidence: 99%