A Robust Chunked Block Lanczos Method with Adaptive Shift Selection for Large-Scale Generalized Eigenvalue Problems in Structural Dynamics

A Robust Chunked Block Lanczos Method with Adaptive Shift Selection for Large-Scale Generalized Eigenvalue Problems in Structural Dynamics | 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 Robust Chunked Block Lanczos Method with Adaptive Shift Selection for Large-Scale Generalized Eigenvalue Problems in Structural Dynamics Anatoly Vershinin This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9248472/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 We present a robust implementation of the shift-invert block Lanczos method for computing hundreds of eigenpairs in large-scale generalized eigenvalue problems arising from finite element analysis of structural dynamics. The proposed algorithm introduces four key innovations: a chunked computation strategy enabling reliable extraction of arbitrarily many eigenpairs without memory overflow, an adaptive shift selection mechanism guaranteeing non-singularity of the shifted operator while optimizing convergence, a dual residual monitoring scheme combining Ritz and true residuals for robust convergence detection, and partial reorthogonalization based on the Simon criterion that reduces computational cost by 33-50%. We demonstrate effectiveness on industrial-scale problems with up to 5 million degrees of freedom. Comprehensive numerical experiments on 18 representative matrix pairs across 45 test configurations demonstrate 100% reliability across problems with rigid body modes, clustered eigenvalues, and high multiplicity. The full validation suite of 58 tests and 1160 robustness runs confirm complete reliability without parameter tuning. Moderate chunking not only bounds memory but dramatically improves performance by up to 89% compared to monolithic computation, enabling large-scale computations that are otherwise infeasible due to memory constraints. Generalized eigenvalue problem Block Lanczos method Shift-invert transformation Structural dynamics Finite element method Large-scale computing 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. 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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-9248472","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":613609457,"identity":"4b016a7d-e98e-44df-aca2-d973eeeb12f7","order_by":0,"name":"Anatoly Vershinin","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2ElEQVRIiWNgGAWjYJACZgYDBh4gzfiAZC3MBiRogQA2CaKUGxzvffy5oOCOjDn7GbOqm20M0fIzEtg+8+DTcua4mfQMg2c8lj05Zrdz2xhyN9xIYJ6NT4vkjDQ2Zh6DwzwGB9LSIFokEpiZ8WqZ/4z5M1jL+WdpxSAt82cQ0MIvwcYgDdZyI/kYM0hLww1CWnjS2KBaHh+WzjknkbvhzMNmxjl4tLCxHwM67M9he4PziY2fc8pscue3Jx9meINHCypgBEcNYwPRGoDgDymKR8EoGAWjYKQAAJTfRKObNLgZAAAAAElFTkSuQmCC","orcid":"","institution":"Lomonosov Moscow State University","correspondingAuthor":true,"prefix":"","firstName":"Anatoly","middleName":"","lastName":"Vershinin","suffix":""}],"badges":[],"createdAt":"2026-03-27 22:08:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9248472/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9248472/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106926725,"identity":"1eba41b4-a8a7-42ca-a036-75f23f1dc759","added_by":"auto","created_at":"2026-04-14 23:24:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":478345,"visible":true,"origin":"","legend":"","description":"","filename":"PaperVershinin.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9248472/v1_covered_fefd0801-bb48-4070-a6d8-f277ac29fc62.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Robust Chunked Block Lanczos Method with Adaptive Shift Selection for Large-Scale Generalized Eigenvalue Problems in Structural Dynamics","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":"Generalized eigenvalue problem, Block Lanczos method, Shift-invert transformation, Structural dynamics, Finite element method, Large-scale computing","lastPublishedDoi":"10.21203/rs.3.rs-9248472/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9248472/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We present a robust implementation of the shift-invert block Lanczos method for computing hundreds of eigenpairs in large-scale generalized eigenvalue problems arising from finite element analysis of structural dynamics. 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