{"title":"Improving the Accuracy of Exponentially Converging Quadratures","authors":"A. A. Belov, V. S. Khokhlachev","doi":"10.1134/s0965542524010020","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Evaluation of one-dimensional integrals arises in many problems in physics and technology. This is most often done using simple quadratures of midpoints, trapezoids and Simpson on a uniform grid. For integrals of periodic functions over the full period, the convergence of these quadratures drastically accelerates and depends on the number of grid steps according to an exponential law. In this paper, new asymptotically accurate estimates of the error of such quadratures are obtained. They take into account the location and multiplicity of the poles of the integrand in the complex plane. A generalization of these estimates is constructed for the case when there is no a priori information about the poles of the integrand. An error extrapolation procedure is described that drastically accelerates the convergence of quadratures.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":"1 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-03-21","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/s0965542524010020","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
Evaluation of one-dimensional integrals arises in many problems in physics and technology. This is most often done using simple quadratures of midpoints, trapezoids and Simpson on a uniform grid. For integrals of periodic functions over the full period, the convergence of these quadratures drastically accelerates and depends on the number of grid steps according to an exponential law. In this paper, new asymptotically accurate estimates of the error of such quadratures are obtained. They take into account the location and multiplicity of the poles of the integrand in the complex plane. A generalization of these estimates is constructed for the case when there is no a priori information about the poles of the integrand. An error extrapolation procedure is described that drastically accelerates the convergence of quadratures.
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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