A low-rank algorithm for computing Lyapunov operator phi-functions within matrix-valued exponential integrators

A low-rank algorithm for computing Lyapunov operator phi-functions within matrix-valued exponential integrators | 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 low-rank algorithm for computing Lyapunov operator phi-functions within matrix-valued exponential integrators Dongping Li, Xiuying Zhang, Hongjiong Tian This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4294394/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 work we introduce a low-rank algorithm designed to compute low-rank approximations of large-scale Lyapunov operator φ-functions. These computations are crucial for the implementation of matrix-valued exponential integrators tailored for large-scale stiff matrix differential equations, where the (approximate) solutions are of low rank. The method is developed using a scaling and recursive procedure, supplemented by a quasi-backward error analysis to determine the optimal parameters. The computational cost of the method primarily arises from the products of sparse matrix and block vectors.Numerical experiments confirm the effectiveness of the proposed method as a foundational tool for matrix-valued exponential integrators in addressing large-scale differential Lyapunov equations and Riccati equations. MSC Classification: 65L05 , 65F10 , 65F30 Low-rank approximation φ-functions Lyapunov operator Matrix-valued exponential integrators 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. 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-4294394","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":305048297,"identity":"e3a9104b-b339-48af-8609-7cdd9eb7fad8","order_by":0,"name":"Dongping Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwElEQVRIiWNgGAWjYBACPmYGA4YPQAZ7A5DgIUYLG1AL4wyQ6gNEa2FgMGDmIU0LO/O2xzY1dvY8EgmMD962McibE3YYW7lxzrHkxB6JBGbDuW0MhjsbCGrhMZPObTiQYC+RwCbN28aQYHCAGC2WDQdADmP/TbwWxoYDjECHsTETqYWt3LAH5Beeh82Sc85JGG4gpIWf//C2Bz9AIcaefPDDmzIbeYK2MICjBgwYG4CEBGH1SFpGwSgYBaNgFOAAALTxMj8pNTiuAAAAAElFTkSuQmCC","orcid":"","institution":"Changchun Normal University","correspondingAuthor":true,"prefix":"","firstName":"Dongping","middleName":"","lastName":"Li","suffix":""},{"id":305048298,"identity":"c804fe5e-e413-4e1b-91d7-9fd9f91e201f","order_by":1,"name":"Xiuying Zhang","email":"","orcid":"","institution":"Changchun Normal University","correspondingAuthor":false,"prefix":"","firstName":"Xiuying","middleName":"","lastName":"Zhang","suffix":""},{"id":305048299,"identity":"600ff9a8-5976-4706-8821-16cf5ceaa99f","order_by":2,"name":"Hongjiong Tian","email":"","orcid":"","institution":"Shanghai Normal University","correspondingAuthor":false,"prefix":"","firstName":"Hongjiong","middleName":"","lastName":"Tian","suffix":""}],"badges":[],"createdAt":"2024-04-19 16:52:46","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4294394/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4294394/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":57899493,"identity":"9098331c-c035-4ec9-872f-9d93e8bf5c19","added_by":"auto","created_at":"2024-06-07 08:29:35","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":398748,"visible":true,"origin":"","legend":"","description":"","filename":"Paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4294394/v1_covered_b8c0ec89-a785-4851-b7ed-69badfeb1b62.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A low-rank algorithm for computing Lyapunov operator phi-functions within matrix-valued exponential integrators","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":"Low-rank approximation, φ-functions, Lyapunov operator, Matrix-valued exponential integrators","lastPublishedDoi":"10.21203/rs.3.rs-4294394/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4294394/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this work we introduce a low-rank algorithm designed to compute low-rank approximations of large-scale Lyapunov operator φ-functions. 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