Convergence properties of Jacobi gradient-based iteration algorithm for the complex conjugate and transpose Sylvester matrix equations

Convergence properties of Jacobi gradient-based iteration algorithm for the complex conjugate and transpose Sylvester matrix equations | 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 Convergence properties of Jacobi gradient-based iteration algorithm for the complex conjugate and transpose Sylvester matrix equations Jiating He, Xuesong Chen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6654794/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 11 Dec, 2025 Read the published version in Numerical Algorithms → Version 1 posted 14 You are reading this latest preprint version Abstract In this paper, inspired by the modified relaxed gradient-based iterative (MRGI) al-gorithm proposed by Huang et al. ( Numer. Algorithms 97(4), 1955–2009 (2024)), a modifiedJacobi gradient iterative (MJGI) algorithm is developed to solve the complex conjugate trans-pose Sylvester matrix equation through updates and modifications. Specifically, we replace theoriginal full matrices by extracting the diagonal parts of the matrices Ai and Bi, while proposinga progressive update mechanism with hybrid historical iterative values. This mechanism dynam-ically integrates the newly computed subblocks with the global historical values Y (l) during theupdating process. By retaining the memory effect of historical information, we enhance the algo-rithmic accuracy. Furthermore, we establish an w-parameterized convergence factor optimizationframework that significantly accelerates the convergence rate. Theoretical analysis, grounded inthe real representation of matrices and Kronecker product operations, establishes the conver-gence conditions of the algorithm and determines explicit optimization ranges for the relaxationfactor and step size parameters. The numerical example demonstrates that the MJGI algorithmoutperforms the MRGI algorithm in terms of iteration counts and CPU running time, especiallyunder high-precision requirements (residual threshold τ ≤ 10−6), with an improvement in it-eration efficiency, defined as the number of iterations required to achieve a specified residualthreshold, by more than 76. 31%. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 11 Dec, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 02 Aug, 2025 Reviews received at journal 11 Jul, 2025 Reviews received at journal 27 Jun, 2025 Reviews received at journal 26 Jun, 2025 Reviews received at journal 13 Jun, 2025 Reviewers agreed at journal 01 Jun, 2025 Reviewers agreed at journal 01 Jun, 2025 Reviewers agreed at journal 22 May, 2025 Reviewers agreed at journal 21 May, 2025 Reviewers agreed at journal 20 May, 2025 Reviewers invited by journal 20 May, 2025 Editor assigned by journal 20 May, 2025 Submission checks completed at journal 19 May, 2025 First submitted to journal 13 May, 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. 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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-6654794","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":460058110,"identity":"5611218f-5cee-4db5-a187-7f5830061444","order_by":0,"name":"Jiating He","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAw0lEQVRIiWNgGAWjYBACA4YDQMggUd/PzHz4AZFaDoO0WDDObGdLMyBSCzOIqmDccJ5HQYIoLeaM5w8eLvglwWx8mAeov8YmmqAWy4bDDIdn9kmwmR3mPfCA4VhabgNBhx0AauHtkeAxO8yXYMDYcJh4LRLGzTwGEsRr4fkhYWDATIIWg8O8DRIJEoeBgZxAlF9uHHz8medPXQJ//+HDDz7U2BDWwiBxgIGBsQ3KSSCoHAT4Qab+IUrpKBgFo2AUjFQAAIDoQyEJFaSqAAAAAElFTkSuQmCC","orcid":"","institution":"Guangdong University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Jiating","middleName":"","lastName":"He","suffix":""},{"id":460058112,"identity":"fe7554a0-7b07-4549-9f8e-6b4f29450358","order_by":1,"name":"Xuesong Chen","email":"","orcid":"","institution":"Guangdong University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Xuesong","middleName":"","lastName":"Chen","suffix":""}],"badges":[],"createdAt":"2025-05-13 11:08:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6654794/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6654794/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02277-5","type":"published","date":"2025-12-11T15:59:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":98245113,"identity":"c345b4ac-7b4c-40a7-b99f-06c70b109e18","added_by":"auto","created_at":"2025-12-15 16:16:39","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":725274,"visible":true,"origin":"","legend":"","description":"","filename":"ConvergencepropertiesofJacobigradientbasediterationalgorithmforthecomplexconjugateandtransposeSylvestermatrixequations.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6654794/v1_covered_da837744-03aa-413b-b593-c67abed84ad7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Convergence properties of Jacobi gradient-based iteration algorithm for the complex conjugate and transpose Sylvester matrix equations","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":"","lastPublishedDoi":"10.21203/rs.3.rs-6654794/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6654794/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this paper, inspired by the modified relaxed gradient-based iterative (MRGI) al-gorithm proposed by Huang et al. 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