Parametric K-Formula: O(n) Closed-Form Solutions for Separable Polynomial Constraints

Parametric K-Formula: O(n) Closed-Form Solutions for Separable Polynomial Constraints | 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 Parametric K-Formula: O(n) Closed-Form Solutions for Separable Polynomial Constraints Rwego T. Serge This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7863157/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 Separable polynomial equations of the form Pn i=1 aixpi i + a0 = 0 appear frequently in engineering applications including trilateration systems, regularized optimization, and geometric constraint satisfaction. Current methods for solving such systems rely primarily on iterative approaches like Newton-Raphson methods, which require O (n 3 ) operations per iteration due to Jacobian matrix computations and can experience convergence difficulties. However, the separable structure of these polynomials presents opportunities for more efficient solution methods that current approaches do not exploit. This study addresses this gap by developing the parametric K-formula (PK-formula), a novel noniterative method that generates closed-form parametric solutions for separable polynomial equations. The approach strategically introduces a tunable parameter k that decomposes the constraint into directly solvable components, enabling solution generation without iterative refinement. We provide complete mathematical derivation with formal proofs, characterize solution domains, and analyze computational complexity. The method’s performance is validated through three industrial applications: real-time positioning systems, regularized optimization, and robotic constraint satisfaction. The PK-formula achieves O(n) computational complexity compared to O (n 3 ) per iteration for Newton-Raphson methods. Experimental validation demonstrates computational speedups ranging from 50× to 120× across the three application domains, with deterministic timing characteristics suitable for real-time systems. The method generates solutions along a specific parametric curve on the constraint manifold where the last n−1 terms contribute equally, providing predictable computational performance. Numerical precision analysis confirms solution accuracy within machine precision limits across all tested scenarios. The results establish that structure-specific approaches can achieve substantial computational advantages for targeted polynomial classes while maintaining mathematical rigor. This work provides a foundation for developing specialized solvers for structured polynomial systems and demonstrates practical utility in applications requiring real-time constraint satisfaction with limited computational resources. polynomial equations noniterative methods computational complexity parametric solutions real-time applications constraint satisfaction 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-7863157","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":530654147,"identity":"6f5d9f1d-ca98-49cd-bc8b-1c913d8fba32","order_by":0,"name":"Rwego T. 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Current methods for solving such systems rely primarily on iterative approaches like Newton-Raphson methods, which require O (n\u003csup\u003e3\u003c/sup\u003e) operations per iteration due to Jacobian matrix computations and can experience convergence difficulties. However, the separable structure of these polynomials presents opportunities for more efficient solution methods that current approaches do not exploit. This study addresses this gap by developing the parametric K-formula (PK-formula), a novel noniterative method that generates closed-form parametric solutions for separable polynomial equations. The approach strategically introduces a tunable parameter k that decomposes the constraint into directly solvable components, enabling solution generation without iterative refinement. We provide complete mathematical derivation with formal proofs, characterize solution domains, and analyze computational complexity. The method’s performance is validated through three industrial applications: real-time positioning systems, regularized optimization, and robotic constraint satisfaction. The PK-formula achieves O(n) computational complexity compared to O (n\u003csup\u003e3\u003c/sup\u003e) per iteration for Newton-Raphson methods. Experimental validation demonstrates computational speedups ranging from 50× to 120× across the three application domains, with deterministic timing characteristics suitable for real-time systems. The method generates solutions along a specific parametric curve on the constraint manifold where the last n−1 terms contribute equally, providing predictable computational performance. Numerical precision analysis confirms solution accuracy within machine precision limits across all tested scenarios. The results establish that structure-specific approaches can achieve substantial computational advantages for targeted polynomial classes while maintaining mathematical rigor. This work provides a foundation for developing specialized solvers for structured polynomial systems and demonstrates practical utility in applications requiring real-time constraint satisfaction with limited computational resources.\u003c/p\u003e","manuscriptTitle":"Parametric K-Formula: O(n) Closed-Form Solutions for Separable Polynomial Constraints","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-05 13:06:25","doi":"10.21203/rs.3.rs-7863157/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"2de1b8fd-919e-4634-abd9-dc03dbdde4b6","owner":[],"postedDate":"November 5th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-11-21T12:53:44+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-05 13:06:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7863157","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7863157","identity":"rs-7863157","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}