Al-Hseno Algorithm: A Robust Derivative-Free Root-Finding Method with Adaptive Multiplicity Estimation, Geometric Damping, and Curvature Correction

Al-Hseno Algorithm: A Robust Derivative-Free Root-Finding Method with Adaptive Multiplicity Estimation, Geometric Damping, and Curvature Correction | 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 Method Article Al-Hseno Algorithm: A Robust Derivative-Free Root-Finding Method with Adaptive Multiplicity Estimation, Geometric Damping, and Curvature Correction Mohammad Alhseno This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9596848/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 introduce Al-Hseno algorithm, a new derivative-free iterative method for solving nonlinear scalar equations of the form f(x) = 0. The proposed algorithm is distinguished by a sophisticated synergy of three adaptive mechanisms that operate seamlessly without requiring any a priori information about the derivative or the multiplicity of the root. An implicit multiplicity estimator, constructed from logarithmic ratios of consecutive function values, dynamically approximates the local order of the root and restores rapid convergence even for multiple roots. Complementing this, an adaptive geometric damping strategy continuously monitors the residual reduction rate and intelligently modulates the step size to suppress oscillations and prevent premature stagnation. Furthermore, a curvaturebased transverse correction, derived from second-order divided differences, compensates for local deviations from linearity and refines the search trajectory. Comprehensive numerical experiments were conducted on a diverse and challenging benchmark suite of 43 nonlinear test functions, encompassing simple roots, multiple roots of varying multiplicity, severe oscillations, and near-singular behavior. The results demonstrate the remarkable robustness of Al-Hseno algorithm, which successfully converged in 41 of the 42 test cases where a real root exists, matching the highest success rate among the tested methods. The algorithm maintains competitive efficiency, typically requiring fewer than 20 function evaluations per problem while attaining residuals at machine precision. These findings position Al-Hseno algorithm as a powerful, reliable, and versatile tool for nonlinear root-finding, particularly in applications where derivative information is unavailable, unreliable, or expensive to compute. Nonlinear equations Derivative-free methods Root-finding algorithms Adaptive multiplicity estimation Geometric correction Numerical robustness Full Text Additional Declarations The authors declare no competing interests. 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-9596848","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Method Article","associatedPublications":[],"authors":[{"id":633488529,"identity":"93042cb0-b289-46c3-9907-40702efcf0bb","order_by":0,"name":"Mohammad Alhseno","email":"data:image/png;base64,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","orcid":"","institution":"University of Aleppo","correspondingAuthor":true,"prefix":"","firstName":"Mohammad","middleName":"","lastName":"Alhseno","suffix":""}],"badges":[],"createdAt":"2026-05-03 01:43:12","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9596848/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9596848/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108466829,"identity":"58fd4a97-f37b-4cbd-a284-cdc239c1e4f8","added_by":"auto","created_at":"2026-05-05 03:56:11","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":328402,"visible":true,"origin":"","legend":"","description":"","filename":"Alhsenomethod5.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9596848/v1_covered_22a8a83c-d88f-4607-8427-dcc2cc978f46.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eAl-Hseno Algorithm: A Robust Derivative-Free Root-Finding Method with Adaptive Multiplicity Estimation, Geometric Damping, and Curvature Correction\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"University of Allepo","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":"Nonlinear equations, Derivative-free methods, Root-finding algorithms, Adaptive multiplicity estimation, Geometric correction, Numerical robustness","lastPublishedDoi":"10.21203/rs.3.rs-9596848/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9596848/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe introduce Al-Hseno algorithm, a new derivative-free iterative method for solving nonlinear scalar equations of the form f(x) = 0. 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