Numerical Study of Spectral Behavior of the Grcar Matrix

Numerical Study of Spectral Behavior of the Grcar Matrix | 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 Numerical Study of Spectral Behavior of the Grcar Matrix Qiang Niu, Wen Jiang, Di Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6576672/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The Grcar matrix is a class of non-symmetric Toeplitz upper Hessenberg matrices with banded structure. In a recent paper [G. Meurant, A note on the Grcar Matrix, Numerical Algorithms, https://doi.org/10.1007/s11075-024-01999-2], the recursive formula for its determinant, LU factorization and asymptotic spectrum have been studied. Inspired by the results, this paper extends the analysis to the characteristic polynomials and inverse representations. Furthermore, we conduct a comprehensive numerical investigation of asymptotic spectra behavior, validating classical Toeplitz matrix theories through visualization of the spectra and pseudospectra distributions. Grcar matrix Spectra Pseudospectra Preconditioner Laurent Polynomial GMRES Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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. 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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-6576672","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":453202071,"identity":"50c254fc-8c61-4ec4-a567-6ad44f369b76","order_by":0,"name":"Qiang Niu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/ElEQVRIie3RsWoCMRzH8d8R8JY/vfWC0mf4g3AODj6CryAIdnEQCuVGQWiX+jAu4hjJcEvAVXFRCk4OgqXo1lxwNedYaL7L/w7y4XIJEAr9wRLhhibEYzBG5UvPT+TEjS8CKbDiBwgrN85Aalc+Rgpx2H/j3ODNkV9PrJHEQ8Z14SG61mo2sCXeDnp2Yxry88jR1PgIsnrqSF85wushi+j9Psl0/GPJinizHDvSqSJNTZk8oSBei9tX0goiJ/RWt4dA0th/MfxCqTmMllMPSbrFXF5y3XkqTMZ53n5OPvqz3dVDygS560CtHFQ+KT8AosuN7qpWhkKh0P/sF4K2UVLNASqdAAAAAElFTkSuQmCC","orcid":"","institution":"Xi’an Jiaotong-Liverpool University","correspondingAuthor":true,"prefix":"","firstName":"Qiang","middleName":"","lastName":"Niu","suffix":""},{"id":453202072,"identity":"2b064ce1-4895-477a-bcfd-9d96a231a34c","order_by":1,"name":"Wen Jiang","email":"","orcid":"","institution":"Xi’an Jiaotong-Liverpool University","correspondingAuthor":false,"prefix":"","firstName":"Wen","middleName":"","lastName":"Jiang","suffix":""},{"id":453202076,"identity":"4a4d0f75-632c-43e2-ae01-d44a582af461","order_by":2,"name":"Di Zhang","email":"","orcid":"","institution":"Xi’an Jiaotong-Liverpool University","correspondingAuthor":false,"prefix":"","firstName":"Di","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2025-05-02 08:23:33","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6576672/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6576672/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":84561159,"identity":"5d524854-b491-47aa-b696-7a1637430ada","added_by":"auto","created_at":"2025-06-13 13:08:28","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2596007,"visible":true,"origin":"","legend":"","description":"","filename":"new.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6576672/v1_covered_9464af50-f15d-433a-8c8a-7ef0df158881.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Numerical Study of Spectral Behavior of the Grcar Matrix","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Grcar matrix, Spectra, Pseudospectra, Preconditioner, Laurent Polynomial, GMRES","lastPublishedDoi":"10.21203/rs.3.rs-6576672/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6576672/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe Grcar matrix is a class of non-symmetric Toeplitz upper Hessenberg matrices with banded structure. 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