On computing the zeros of a class of Sobolev orthogonal polynomials

On computing the zeros of a class of Sobolev orthogonal polynomials | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article On computing the zeros of a class of Sobolev orthogonal polynomials Nicola Mastronardi, Marc Van Barel, Raf Vandebril, Paul Van Dooren This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6314872/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 31 May, 2025 Read the published version in Numerical Algorithms → Version 1 posted 7 You are reading this latest preprint version Abstract A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre polynomials. The theoretical study of the asymptotic distribution of the spectrum of these polynomials is an active research topic. In this article we do not contribute to the theory, but provide a practical method to contribute to further and better understanding of the asymptotic behavior. The polynomials under consideration fit into the class of Sobolev orthogonal polynomials, satisfying a four--term recurrence relation. This allows computing the roots via a generalized eigenvalue problem. After condition enhancing similarity transformations, the problem is transformed into the computation of the eigenvalues of a comrade matrix, which is a symmetric tridiagonal modified by a rank--one matrix. The eigenvalues are then retrieved by relying on an existing structured rank based fast algorithm. Numerical examples are reported studying the accuracy, stability and conforming the efficiency for various parameter settings of the proposed approach. AMS Classification: 33C20, 65F15, 65F35 Sobolev orthogonal polynomials zeros of polynomials generalized eigenvalue problem comrade matrices Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 31 May, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 08 May, 2025 Reviews received at journal 08 May, 2025 Reviewers agreed at journal 31 Mar, 2025 Reviewers invited by journal 28 Mar, 2025 Editor assigned by journal 28 Mar, 2025 Submission checks completed at journal 28 Mar, 2025 First submitted to journal 26 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6314872","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":439619473,"identity":"daf8ae66-f647-4116-92f8-93dbfa30bdcf","order_by":0,"name":"Nicola Mastronardi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVRIiWNgGAWjYDACZiB+gCJymPkAiOLBqyUBzgJrYQPzeXDrwdBygMeAAZ815uzMDx8k1DBE8/efP/bh4w67fL7jPB8/F+5hkLHHocWymc3YIOEYQ+6MG8nMM2eeSbaceZh3s/SMZ7gdZnCYh00igY0ht+EGMzMzbxuzgcFh3m3MPAcIafnHkDv//GGQlnqgFp5nhLUktjHkbjiQDNJyGKSFjYAWoF8S+yRyN95INmac2XbcQBIoIs1zQIIHqA27lvOHHz748M0md975g48ZPrZVG/ABRT7zHLCxZ2/AYQ0ESBAhMgpGwSgYBaOAeAAAnedPwi1DmZcAAAAASUVORK5CYII=","orcid":"","institution":"National Research Council","correspondingAuthor":true,"prefix":"","firstName":"Nicola","middleName":"","lastName":"Mastronardi","suffix":""},{"id":439619475,"identity":"1b9c87c4-6cb4-469c-add9-18b6b48bb6e9","order_by":1,"name":"Marc Van Barel","email":"","orcid":"","institution":"KU Leuven","correspondingAuthor":false,"prefix":"","firstName":"Marc","middleName":"Van","lastName":"Barel","suffix":""},{"id":439619477,"identity":"a65c5b9e-ab9a-49da-8a27-bdb47ddce450","order_by":2,"name":"Raf Vandebril","email":"","orcid":"","institution":"KU Leuven","correspondingAuthor":false,"prefix":"","firstName":"Raf","middleName":"","lastName":"Vandebril","suffix":""},{"id":439619479,"identity":"e2023f92-8593-4702-b6b3-27a2988a8520","order_by":3,"name":"Paul Van Dooren","email":"","orcid":"","institution":"Université Catholique de Louvain","correspondingAuthor":false,"prefix":"","firstName":"Paul","middleName":"Van","lastName":"Dooren","suffix":""}],"badges":[],"createdAt":"2025-03-26 18:53:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6314872/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6314872/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02106-9","type":"published","date":"2025-05-31T15:57:31+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":83782888,"identity":"55d83d41-b843-44a8-ae74-0817a4a836b8","added_by":"auto","created_at":"2025-06-02 16:08:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":693027,"visible":true,"origin":"","legend":"","description":"","filename":"ComputingthezerosofaclassofSobolevorthogonalpolynomials.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6314872/v1_covered_0e29849f-0913-4af7-adc6-b80e0981984a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On computing the zeros of a class of Sobolev orthogonal polynomials","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Sobolev orthogonal polynomials, zeros of polynomials, generalized eigenvalue problem, comrade matrices","lastPublishedDoi":"10.21203/rs.3.rs-6314872/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6314872/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eA fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. 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