{"title":"P-Factor Interpolation of Solutions of an Equation with a Degenerate Function","authors":"Yu. G. Evtushenko, A. A. Tret’yakov","doi":"10.1134/S1064562424601689","DOIUrl":null,"url":null,"abstract":"<p>The paper considers a new method for interpolation of nonlinear functions on an interval, the so-called <i>p</i>-factor interpolation method. By using a Newton interpolation polynomial as an example, it is shown that, in the case of degeneration of the approximated function <span>\\(f(x)\\)</span> in the solution, classical interpolation does not provide the necessary accuracy for finding an approximate solution to the equation <span>\\(f(x) = 0\\)</span>, in contrast to the nondegenerate regular case. In turn, the use of <i>p</i>-factor interpolation polynomials for approximating functions in order to obtain the desired approximate solution to the equation provides the necessary order of accuracy in the argument during the calculations. The results are based on constructions of <i>p</i>-regularity theory and the apparatus of <i>p</i>-factor operators, which are effectively used in the study of degenerate mappings.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"110 3","pages":"451 - 456"},"PeriodicalIF":0.5000,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Doklady Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S1064562424601689","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The paper considers a new method for interpolation of nonlinear functions on an interval, the so-called p-factor interpolation method. By using a Newton interpolation polynomial as an example, it is shown that, in the case of degeneration of the approximated function \(f(x)\) in the solution, classical interpolation does not provide the necessary accuracy for finding an approximate solution to the equation \(f(x) = 0\), in contrast to the nondegenerate regular case. In turn, the use of p-factor interpolation polynomials for approximating functions in order to obtain the desired approximate solution to the equation provides the necessary order of accuracy in the argument during the calculations. The results are based on constructions of p-regularity theory and the apparatus of p-factor operators, which are effectively used in the study of degenerate mappings.
期刊介绍：
Doklady Mathematics is a journal of the Presidium of the Russian Academy of Sciences. It contains English translations of papers published in Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences), which was founded in 1933 and is published 36 times a year. Doklady Mathematics includes the materials from the following areas: mathematics, mathematical physics, computer science, control theory, and computers. It publishes brief scientific reports on previously unpublished significant new research in mathematics and its applications. The main contributors to the journal are Members of the RAS, Corresponding Members of the RAS, and scientists from the former Soviet Union and other foreign countries. Among the contributors are the outstanding Russian mathematicians.
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