{"title":"Exact Formula for Solving a Degenerate System of Quadratic Equations","authors":"Yu. G. Evtushenko, A. A. Tret’yakov","doi":"10.1134/s0965542524030072","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper is devoted to the solution of a nonlinear system of equations <span>\\(F(x{{) = 0}_{n}}\\)</span>, where <span>\\(F\\)</span> is a quadratic mapping acting from <span>\\({{\\mathbb{R}}^{n}}\\)</span> to <span>\\({{\\mathbb{R}}^{n}}\\)</span>. The derivative <span>\\(F{\\kern 1pt} '\\)</span> is assumed to be degenerate at the solution point, which is a major characteristic property of nonlinearity of the mapping. Based on constructions of the <i>p</i>-regularity theory, a 2-factor method is proposed for solving the system of equations, which converges at a quadratic rate. Moreover, an exact formula is obtained for solving this quadratic system of equations in the case of a 2-regular mapping <span>\\(F(x)\\)</span>.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Mathematics and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0965542524030072","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The paper is devoted to the solution of a nonlinear system of equations \(F(x{{) = 0}_{n}}\), where \(F\) is a quadratic mapping acting from \({{\mathbb{R}}^{n}}\) to \({{\mathbb{R}}^{n}}\). The derivative \(F{\kern 1pt} '\) is assumed to be degenerate at the solution point, which is a major characteristic property of nonlinearity of the mapping. Based on constructions of the p-regularity theory, a 2-factor method is proposed for solving the system of equations, which converges at a quadratic rate. Moreover, an exact formula is obtained for solving this quadratic system of equations in the case of a 2-regular mapping \(F(x)\).
期刊介绍：
Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.
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