A block upper triangular preconditioner for block three-by-three linear systems arising form the large indefinite least squares problem

A block upper triangular preconditioner for block three-by-three linear systems arising form the large indefinite least squares problem | 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 A block upper triangular preconditioner for block three-by-three linear systems arising form the large indefinite least squares problem Jun Li, Kailiang Xin, Lingsheng Meng This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4557127/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 In this paper, to solve the large indefinite least squares problem, we first rewrite the normal equation of it as the sparse block three-by-three linear systems with non-singular diagonal blocks, then GMRES method is used to solve the linear systems. In order to make GMRES method converge quickly, a block upper triangular preconditioner is deduced. Theoretically, the iteration method under the preconditioner is unconditional convergent. Furthermore, the all eigenvalues of the preconditioned matrix are real number and located in a positive interval. Numerical results are provided to support the validity of the theoretical results and demonstrate the effectiveness of the studied precondition. Mathematics Subject Classification (2010) 65F10 · 65F20 Indefinite least squares problem Block three-by-three linear systems Preconditioning GMRES method 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-4557127","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":319287100,"identity":"e476592a-8fb0-40e7-a306-4ba56249360e","order_by":0,"name":"Jun Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVRIie2RrQ7CMBRG79JkmBHslhB4hS5LpniYWzM1CAmmAtEFsgkg2D0GEkeTJVPFI5nH4HBQkAhaieiRN9/J/QNwOP4QT4AnkU5G3r4orsiXVgpI5FlC6qahV9XadZKgGtYTWRZ1a2KOk3oqZVcS7AuVciZ8GFQb/D1YPUPJSn8WrbbphR2HEKrzwaDkVCvBItZdLkz5QMOplRKyk8zTOSuJpYKKskKvD3bK9qYVjomnjxyiagPjLnGVx92DPj+vvD/4cjSodgZFfBWCn/E3Y2PC4XA4HC8u31OdmQMdpAAAAABJRU5ErkJggg==","orcid":"","institution":"Lanzhou University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Jun","middleName":"","lastName":"Li","suffix":""},{"id":319287101,"identity":"8e1da73a-0445-42b8-8b06-d1e0fd75247d","order_by":1,"name":"Kailiang Xin","email":"","orcid":"","institution":"Northwest Normal University","correspondingAuthor":false,"prefix":"","firstName":"Kailiang","middleName":"","lastName":"Xin","suffix":""},{"id":319287102,"identity":"85010306-c293-4aa2-a909-1c07b4c49fcb","order_by":2,"name":"Lingsheng Meng","email":"","orcid":"","institution":"Northwest Normal University","correspondingAuthor":false,"prefix":"","firstName":"Lingsheng","middleName":"","lastName":"Meng","suffix":""}],"badges":[],"createdAt":"2024-06-10 09:36:56","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4557127/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4557127/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":63355298,"identity":"dd723e63-c1a5-422a-acbc-2dd548c9baf2","added_by":"auto","created_at":"2024-08-27 09:09:06","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":225810,"visible":true,"origin":"","legend":"","description":"","filename":"springer.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4557127/v1_covered_77050efc-42e4-4db0-8c8f-dda2a314d7e1.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A block upper triangular preconditioner for block three-by-three linear systems arising form the large indefinite least squares problem","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"Indefinite least squares problem, Block three-by-three linear systems, Preconditioning, GMRES method","lastPublishedDoi":"10.21203/rs.3.rs-4557127/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4557127/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, to solve the large indefinite least squares problem, we first rewrite the normal equation of it as the sparse block three-by-three linear systems with non-singular diagonal blocks, then GMRES method is used to solve the linear systems. In order to make GMRES method converge quickly, a block upper triangular preconditioner is deduced. Theoretically, the iteration method under the preconditioner is unconditional convergent. Furthermore, the all eigenvalues of the preconditioned matrix are real number and located in a positive interval. Numerical results are provided to support the validity of the theoretical results and demonstrate the effectiveness of the studied precondition.\u003c/p\u003e\n\u003cp\u003eMathematics Subject Classification (2010) 65F10 · 65F20\u003c/p\u003e","manuscriptTitle":"A block upper triangular preconditioner for block three-by-three linear systems arising form the large indefinite least squares problem","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-06-27 02:32:45","doi":"10.21203/rs.3.rs-4557127/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"535df8fc-7fc1-447e-a54a-ace6cc7af45e","owner":[],"postedDate":"June 27th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-08-27T09:08:57+00:00","versionOfRecord":[],"versionCreatedAt":"2024-06-27 02:32:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4557127","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4557127","identity":"rs-4557127","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}