An Adaptive Quantile-Based Block Kaczmarz Method for Nonlinear Equations

An Adaptive Quantile-Based Block Kaczmarz Method for Nonlinear 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 An Adaptive Quantile-Based Block Kaczmarz Method for Nonlinear Equations Peng Chen, Xiao-Xia Guo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8886967/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 This paper introduces an adaptive block Kaczmarz algorithm for solving large-scale systems of nonlinear equations. The core innovation is a quantile-based criterion for dynamically selecting the working rows at each iteration. By focusing on equations associated with the largest residuals, the method efficiently projects the iterate onto the corresponding hyperplanes, circumventing the need for computationally expensive submatrix pseudoinverses. This approach offers enhanced flexibility in block size selection compared to existing block nonlinear Kaczmarz methods. We provide a rigorous convergence analysis, demonstrating that with an appropriate quantile parameter, the upper bound on the convergence rate is superior to that of related Kaczmarz-type algorithms. Numerical experiments on a variety of test problems confirm that the proposed method significantly outperforms its counterparts in both iteration count and computational time. Mathematics Subject Classification: 65H10, 65F10, 65Y20 Block Kaczmarz algorithm Quantile-based Nonlinear equations Adaptive 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-8886967","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":597114156,"identity":"d0d206e8-98e0-4b49-92b3-09844bb014fa","order_by":0,"name":"Peng Chen","email":"","orcid":"","institution":"Ocean University of China","correspondingAuthor":false,"prefix":"","firstName":"Peng","middleName":"","lastName":"Chen","suffix":""},{"id":597114157,"identity":"f871f8ef-0250-40a7-8d6f-85e6ea4d61f0","order_by":1,"name":"Xiao-Xia Guo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4klEQVRIiWNgGAWjYLCCBCBmY2A+AOEdIF4LWwIJWiCAx4A4LfIRycckHtTYJPaxn/n4mKeGQY7vRgLj5wI8WgxvpCUbJBxLM2bjyd1szHOMwVjyRgKz9Ax8WmbkGD5IYDssx8aQu02at4EhccONBDZmHvxaDA4k/DvMw8b/5hlISz1BLfISQFsS24C2SOSwgbQkGBDSYsDzLNkgsQ/oF4lnxoZzjkkYzjzzsFkary3tycckf3yzSZzfn/zwwZsaG3m+48kHP+O15QAqXwKIGRvwaADagl96FIyCUTAKRgEQAACgjkZszsMPDQAAAABJRU5ErkJggg==","orcid":"","institution":"Ocean University of China","correspondingAuthor":true,"prefix":"","firstName":"Xiao-Xia","middleName":"","lastName":"Guo","suffix":""}],"badges":[],"createdAt":"2026-02-15 15:24:00","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8886967/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8886967/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108970495,"identity":"49c8e2fb-3ebd-4d87-a54a-3388bc7de361","added_by":"auto","created_at":"2026-05-11 10:16:31","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1090586,"visible":true,"origin":"","legend":"","description":"","filename":"NQBKtex.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8886967/v1_covered_2dcd79fe-ac15-4589-9f23-954f02d59a53.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An Adaptive Quantile-Based Block Kaczmarz Method for Nonlinear Equations","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":"Block Kaczmarz algorithm, Quantile-based, Nonlinear equations, Adaptive","lastPublishedDoi":"10.21203/rs.3.rs-8886967/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8886967/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper introduces an adaptive block Kaczmarz algorithm for solving large-scale systems of nonlinear equations. 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