The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929589375205376 |
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| author | Provoost, Evert Michiels, Wim |
| author_facet | Provoost, Evert Michiels, Wim |
| contents | We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Padé approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the non-zero collocation points are chosen as the zeroes of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the $H^2$-norm, where, under mild conditions, super-geometric convergence is observed and, for a special case, super convergence is proved; both significantly faster than the algebraic convergence reported in previous work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited Provoost, Evert Michiels, Wim Numerical Analysis 65L03, 34K06, 15A24 We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Padé approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the non-zero collocation points are chosen as the zeroes of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the $H^2$-norm, where, under mild conditions, super-geometric convergence is observed and, for a special case, super convergence is proved; both significantly faster than the algebraic convergence reported in previous work. |
| title | The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited |
| topic | Numerical Analysis 65L03, 34K06, 15A24 |
| url | https://arxiv.org/abs/2403.03895 |