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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| Vydáno v: | Applied mathematics and computation Ročník 466; s. 128483 |
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| Jazyk: | angličtina |
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01.04.2024
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| ISSN: | 0096-3003, 1873-5649 |
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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 128483 |
| Author | Vassalos, P. Serra-Capizzano, S. Bogoya, M. |
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| References | Serra-Capizzano (10.1016/j.amc.2023.128483_br0190) 1997; 267 Dumbser (10.1016/j.amc.2023.128483_br0360) 2018; 25 Tyrtyshnikov (10.1016/j.amc.2023.128483_br0240) 1998; 270 Serra-Capizzano (10.1016/j.amc.2023.128483_br0200) 1999; 20 Garoni (10.1016/j.amc.2023.128483_br0350) 2018; 7 Bogoya (10.1016/j.amc.2023.128483_br0090) 2017; 208 Tyrtyshnikov (10.1016/j.amc.2023.128483_br0320) 1996; 232 Garoni (10.1016/j.amc.2023.128483_br0380) 2019; 26 Bogoya (10.1016/j.amc.2023.128483_br0070) 2016; 493 Böttcher (10.1016/j.amc.2023.128483_br0120) 1999 Serra-Capizzano (10.1016/j.amc.2023.128483_br0180) 1994; 34 Avram (10.1016/j.amc.2023.128483_br0300) 1988; 79 Ekström (10.1016/j.amc.2023.128483_br0050) 2019; 80 Parter (10.1016/j.amc.2023.128483_br0310) 1986; 80 Bogoya (10.1016/j.amc.2023.128483_br0100) Bogoya (10.1016/j.amc.2023.128483_br0330) Serra-Capizzano (10.1016/j.amc.2023.128483_br0210) 1999; 39 Tilli (10.1016/j.amc.2023.128483_br0250) 1998; 45 Ekström (10.1016/j.amc.2023.128483_br0010) 2018; 27 Grenander (10.1016/j.amc.2023.128483_br0230) 1984 Garoni (10.1016/j.amc.2023.128483_br0150) 2017 Stoer (10.1016/j.amc.2023.128483_br0270) 2010 Garoni (10.1016/j.amc.2023.128483_br0160) 2018 Bogoya (10.1016/j.amc.2023.128483_br0290) 2015; 196 Di-Benedetto (10.1016/j.amc.2023.128483_br0170) 1993; 25 Barrera (10.1016/j.amc.2023.128483_br0340) 2017; 259 Garoni (10.1016/j.amc.2023.128483_br0370) 2015; 36 Bogoya (10.1016/j.amc.2023.128483_br0080) 2017; 259 Barbarino (10.1016/j.amc.2023.128483_br0130) 2020; 53 Bogoya (10.1016/j.amc.2023.128483_br0260) 2023; 271 Serra-Capizzano (10.1016/j.amc.2023.128483_br0220) 1998; 282 Davis (10.1016/j.amc.2023.128483_br0280) 1975 Ekström (10.1016/j.amc.2023.128483_br0030) 2018; 25 Ahmad (10.1016/j.amc.2023.128483_br0020) 2018; 78 Ekström (10.1016/j.amc.2023.128483_br0040) 2018; 58 Bogoya (10.1016/j.amc.2023.128483_br0110) 2022; 91 Barbarino (10.1016/j.amc.2023.128483_br0140) 2020; 53 Bogoya (10.1016/j.amc.2023.128483_br0060) 2015; 422 |
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| SubjectTerms | Asymptotic expansion Beräkningsvetenskap med inriktning mot numerisk analys Numerical algorithm Preconditioned matrix Scientific Computing with specialization in Numerical Analysis Spectra Toeplitz matrix |
| Title | Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory: The preconditioned setting |
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