High-Performance Damped Traub-Type Iterative Scheme for Nonlinear Problems

High-Performance Damped Traub-Type Iterative Scheme for Nonlinear Problems | 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 High-Performance Damped Traub-Type Iterative Scheme for Nonlinear Problems Alicia Cordero, Renso V. Rojas-Hiciano, Juan R. Torregrosa This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9145523/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 present a new family of high-performance schemes for solving non-linear systems. The method is simple and offers a very low computational cost. The convergence analysis shows that the order reaches twice the number of steps. We focus on a new eighth-order method that requires to solve only four linear systems per iteration. These systems share the same Jacobian matrix and a simple scalar weight function. Efficiency studies show that our method outperforms other optimal eighth-order and highly efficient ninth-order schemes in the literature. A dynamical analysis confirms its superior stability, showing large and connected convergence regions. Monte Carlo tests on the Bratu problem prove that the new scheme can exceed the stability of the optimal second-order Newton's method, introducing a new robustness test by using uncertainty vs. non-linearity. High-precision numerical tests on an elastic string problem validate the theoretical results. This work enables the practical use of very high-order schemes with minimal computational effort. Systems of non-linear equations Iterative methods Damped Traub-type Vectorial optimal eighth-order Global stability Bratu problem Elastic string problem 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. 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