{"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/finding-the-nearest-o-stable-pencil-with","title":"Finding the nearest $Ω$-stable pencil with Riemannian optimization","arxiv_id":"2501.16876","date":"2025-01-28","proceeding":null,"authors":["Vanni Noferini","Lauri Nyman"],"abstract":"This paper considers the problem of finding the nearest $\\Omega$-stable pencil to a given square pencil $A+xB \\in \\mathbb{C}^{n \\times n}$, where a pencil is called $\\Omega$-stable if it is regular and all of its eigenvalues belong to the closed set $\\Omega$. We propose a new method, based on the Schur form of a matrix pair and Riemannian optimization over the manifold $U(n) \\times U(n)$, that is, the Cartesian product of the unitary group with itself. While the developed theory holds for any closed set $\\Omega$, we focus on two cases that are the most common in applications: Hurwitz stability and Schur stability. For these cases, we develop publicly available efficient implementations. Numerical experiments show that the resulting algorithm outperforms existing methods.","url_abs":"https://arxiv.org/abs/2501.16876v1","url_pdf":"https://arxiv.org/pdf/2501.16876v1.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":"finding-the-nearest-o-stable-pencil-with","repo_url":"https://github.com/nymanlauri/nearest-stable-pencil","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}