{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-more-direct-and-better-variant-of-new-q","title":"A more direct and better variant of New Q-Newton's method Backtracking for m equations in m variables","arxiv_id":"2110.07403","date":"2021-10-14","proceeding":null,"authors":["Tuyen Trung Truong"],"abstract":"In this paper we apply the ideas of New Q-Newton's method directly to a system of equations, utilising the specialties of the cost function $f=||F||^2$, where $F=(f_1,\\ldots ,f_m)$. The first algorithm proposed here is a modification of Levenberg-Marquardt algorithm, where we prove some new results on global convergence and avoidance of saddle points. The second algorithm proposed here is a modification of New Q-Newton's method Backtracking, where we use the operator $\\nabla ^2f(x)+\\delta ||F(x)||^{\\tau}$ instead of $\\nabla ^2f(x)+\\delta ||\\nabla f(x)||^{\\tau}$. This new version is more suitable than New Q-Newton's method Backtracking itself, while currently has better avoidance of saddle points guarantee than Levenberg-Marquardt algorithms. Also, a general scheme for second order methods for solving systems of equations is proposed. We will also discuss a way to avoid that the limit of the constructed sequence is a solution of $H(x)^{\\intercal}F(x)=0$ but not of $F(x)=0$.","url_abs":"https://arxiv.org/abs/2110.07403v2","url_pdf":"https://arxiv.org/pdf/2110.07403v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-more-direct-and-better-variant-of-new-q","repo_url":"https://github.com/tuyenttMathOslo/NewQNewtonMethodBacktrackingForSystemEquations","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}