Third Order Methods for Multiple Roots of Nonlinear Equations  with Applications

Third Order Methods for Multiple Roots of Nonlinear Equations with Applications | 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 Third Order Methods for Multiple Roots of Nonlinear Equations with Applications Changbum Chun, Beny Neta This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8339198/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract Several new third-order iterative methods for finding multiple roots of nonlinear equations are developed and systematically compared with well-known existing schemes. These methods leverage knowledge of the root multiplicity and include generalizations of Newton, Halley, and Euler-Cauchy-type methods. Both qualitative and quantitative analyses are conducted: basins of attraction are employed to visualize convergence behavior in the complex plane, and metrics such as iteration count, divergence rate, and computational efficiency are systematically recorded. Among the 29 constructed methods, nine demonstrate competitive performance with classical approaches. One method consistently outperforms classical approaches, achieving the shortest average CPU time across all tested examples. The results show that careful parameter selection within the general iterative framework leads to robust and efficient methods for multiple roots, with potential applications in engineering problems such as supersonic flow and complex fluid modeling as demonstrated in the last two examples from chemistry. MSC Classification: 65H05 , 65B99 , 65Y20 Iterative method Basins of attraction Multiple roots Nonlinear equations Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 13 Mar, 2026 Reviews received at journal 01 Mar, 2026 Reviewers agreed at journal 24 Jan, 2026 Reviewers invited by journal 22 Jan, 2026 Editor assigned by journal 12 Dec, 2025 Submission checks completed at journal 12 Dec, 2025 First submitted to journal 11 Dec, 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. 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-8339198","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":579688741,"identity":"82bf0380-7355-484d-a9a0-e7da0a204eb5","order_by":0,"name":"Changbum Chun","email":"","orcid":"","institution":"Sungkyunkwan University","correspondingAuthor":false,"prefix":"","firstName":"Changbum","middleName":"","lastName":"Chun","suffix":""},{"id":579688743,"identity":"dd9623f9-03c3-4541-9787-db985b3c7898","order_by":1,"name":"Beny Neta","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVRIiWNgGAWjYBACAwhlA8SMjQeI13KAIQ2kpYEkLYcZIDQxwFwi9+DjDzXn7da2HwbaUmMTTVCL5Yy8ZIMDx24nbzuTCNRyLC23gaDDbuSYSRxgu51sdgCohbHhMFFazH8c+Hcu2ez8Q+K1mDEcbDtgZ3aDWFsse94YS5ztS04wuwG0JYEYv5iz5xh+qPhmZ292Pv3hgw81NoS1MAgkgKlEsMoEgspBgP8AmLInSvEoGAWjYBSMTAAArDlL7ek4bbUAAAAASUVORK5CYII=","orcid":"","institution":"Naval Postgraduate School","correspondingAuthor":true,"prefix":"","firstName":"Beny","middleName":"","lastName":"Neta","suffix":""}],"badges":[],"createdAt":"2025-12-11 17:38:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8339198/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8339198/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101213974,"identity":"32eae0bc-7328-4372-bc48-89926ba535e3","added_by":"auto","created_at":"2026-01-27 10:32:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3209934,"visible":true,"origin":"","legend":"","description":"","filename":"MultipleJMC.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8339198/v1_covered_f75b54f4-0f19-4bd7-825d-accf08fa9ec0.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Third Order Methods for Multiple Roots of Nonlinear Equations with Applications","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":"journal-of-mathematical-chemistry","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jomc","sideBox":"Learn more about [Journal of Mathematical Chemistry](http://link.springer.com/journal/10910)","snPcode":"10910","submissionUrl":"https://submission.nature.com/new-submission/10910/3","title":"Journal of Mathematical Chemistry","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Iterative method, Basins of attraction, Multiple roots, Nonlinear equations","lastPublishedDoi":"10.21203/rs.3.rs-8339198/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8339198/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSeveral new third-order iterative methods for finding multiple roots of nonlinear equations are developed and systematically compared with well-known existing schemes. 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