Finding roots of complex analytic functions via generalized colleague matrices

[2307.14494] Finding roots of complex analytic functions via generalized colleague matrices Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Numerical Analysis arXiv:2307.14494 (math) [Submitted on 26 Jul 2023 ( v1 ), last revised 9 Apr 2026 (this version, v4)] Title: Finding roots of complex analytic functions via generalized colleague matrices Authors: Hanwen Zhang , Vladimir Rokhlin View a PDF of the paper titled Finding roots of complex analytic functions via generalized colleague matrices, by Hanwen Zhang and 1 other authors View PDF HTML (experimental) Abstract: We present a scheme for finding all roots of an analytic function in a square domain in the complex plane. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on the real line, by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called ''colleague matrices''. Our extension of the classical approach is based on several observations that enable the construction of polynomial bases in compact domains that satisfy three-term recurrences and are reasonably well-conditioned. This class of polynomial bases gives rise to ''generalized colleague matrices'', whose eigenvalues are roots of functions expressed in these bases. In this paper, we also introduce a special-purpose QR algorithm for finding the eigenvalues of generalized colleague matrices, which is a straightforward extension of the recently introduced componentwise stable QR algorithm for the classical cases (See [Serkh]). The performance of the schemes is illustrated with several numerical examples. Subjects: Numerical Analysis (math.NA) Report number: YALEU/DCS/TR-1566 Cite as: arXiv:2307.14494 [math.NA] (or arXiv:2307.14494v4 [math.NA] for this version) https://doi.org/10.48550/arXiv.2307.14494 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Adv Comput Math 50, 71 (2024) Related DOI : https://doi.org/10.1007/s10444-024-10174-z Focus to learn more DOI(s) linking to related resources Submission history From: Hanwen Zhang [ view email ] [v1] Wed, 26 Jul 2023 20:40:21 UTC (881 KB) [v2] Mon, 16 Oct 2023 20:35:57 UTC (514 KB) [v3] Fri, 11 Oct 2024 04:27:32 UTC (517 KB) [v4] Thu, 9 Apr 2026 20:17:35 UTC (515 KB) Full-text links: Access Paper: View a PDF of the paper titled Finding roots of complex analytic functions via generalized colleague matrices, by Hanwen Zhang and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2023-07 Change to browse by: cs cs.NA math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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