Efficient Higher-Order Iterative Schemes for Enhanced Convergence and Dynamical Analysis in Real-World Applications.

Efficient Higher-Order Iterative Schemes for Enhanced Convergence and Dynamical Analysis in Real-World 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 Efficient Higher-Order Iterative Schemes for Enhanced Convergence and Dynamical Analysis in Real-World Applications. Muhammad Raza, Mashood Ul haq, Daanish Toheed, Najma Abdul Rehman This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7439311/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 fourth, fifth, sixth and seventh-order iterative schemes by using a Newton-like method and weight functions. We utilize the Newton method as the initial step, followed by Newton-like methods for the second and third steps. The proposed schemes are numerically tested and analyzed for convergence. The intention is to demonstrate the efficiency and validity of the proposed fourth, fifth, sixth and seventh-order methods. Furthermore, we examine the dynamical behavior by discussing the basin of attraction. The Basin of attraction shows that our schemes produce bigger regions as compared to some existing methods, which makes them a good competitor. The order of convergence of these methods is confirmed theoretically and their computational performance is examined when applied to real-world nonlinear problems encountered in a various field such as blood rheology, fluid mechanics, economics, chemical engineering, and quantum mechanics. Math Subject Classifications 2020: 65H04, 65H05, 90C39. Pure Mathematics Computational Mathematics Nonlinear Equations Iterative Methods Order of Convergence Dynamical Behavior Basin of Attraction Stability 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-7439311","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":504496011,"identity":"acb6840f-253f-4b17-a85c-5c08925726b9","order_by":0,"name":"Muhammad Raza","email":"","orcid":"https://orcid.org/0000-0002-1026-0856","institution":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus","correspondingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"","lastName":"Raza","suffix":""},{"id":504496012,"identity":"2f3253ae-3e8a-4af0-875e-411862918cea","order_by":1,"name":"Mashood Ul haq","email":"","orcid":"","institution":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus","correspondingAuthor":false,"prefix":"","firstName":"Mashood","middleName":"Ul","lastName":"haq","suffix":""},{"id":504496139,"identity":"660836d9-bba1-4b5f-9041-7d101b43d886","order_by":2,"name":"Daanish Toheed","email":"","orcid":"","institution":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus","correspondingAuthor":false,"prefix":"","firstName":"Daanish","middleName":"","lastName":"Toheed","suffix":""},{"id":504496204,"identity":"da8a6e83-f71b-46e1-9845-ef1b135d9853","order_by":3,"name":"Najma Abdul Rehman","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVRIiWNgGAWjYHCCBDDJLwGmJGSI1yI5g4GxAaiFh3i7DG6AtTAQ1mLOfuDphh8MdvnGt5uPP7pRY8HDwH746AZ8Wix7EtJu9jAkW267cyyxOecY0GE8aWk38LrnQELaDR4GZgOzGzmGzTlsQC0SPGb4tZx/kHbzD0O9gfEMkJZ/xGi5kZB2m4fhsIGBBFBLbhtRWh6k3ZZhOG4gcSMtcXZunwQPG0G/nM9Ju/mGodqAf0bygc853+rk+NkPH8OrBRgRCQyM/5D4bPiVgwD7AcJqRsEoGAWjYGQDAJrER7z3AihlAAAAAElFTkSuQmCC","orcid":"","institution":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus","correspondingAuthor":true,"prefix":"","firstName":"Najma","middleName":"Abdul","lastName":"Rehman","suffix":""}],"badges":[],"createdAt":"2025-08-23 07:13:10","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-7439311/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7439311/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89869605,"identity":"ab285dbc-6555-4478-a7db-7ee0f3445a7a","added_by":"auto","created_at":"2025-08-26 02:07:24","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1672192,"visible":true,"origin":"","legend":"","description":"","filename":"Paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7439311/v1_covered_f3959acd-35bd-4698-93f8-1a6befa526cb.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eEfficient Higher-Order Iterative Schemes for Enhanced Convergence and Dynamical Analysis in Real-World Applications.\u003c/strong\u003e\u003c/p\u003e","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":"Nonlinear Equations, Iterative Methods, Order of Convergence, Dynamical Behavior, Basin of Attraction, Stability","lastPublishedDoi":"10.21203/rs.3.rs-7439311/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7439311/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper introduces fourth, fifth, sixth and seventh-order iterative schemes by using a Newton-like method and weight functions. 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