{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-lanczos-tau-framework-for-time-delay","title":"The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited","arxiv_id":"2403.03895","date":"2024-03-06","proceeding":null,"authors":["Evert Provoost","Wim Michiels"],"abstract":"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 Pad\\'e 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.","url_abs":"https://arxiv.org/abs/2403.03895v2","url_pdf":"https://arxiv.org/pdf/2403.03895v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-lanczos-tau-framework-for-time-delay","repo_url":"https://zenodo.org/record/12088674","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}