{"title":"求积分公式的准确性","authors":"L. Trefethen","doi":"10.1137/20m1389522","DOIUrl":null,"url":null,"abstract":"The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases: Newton–Cotes, Clenshaw–Curtis, Gauss–Legendre, and Gauss–Hermite quadrature. Three further examples are mentioned more briefly.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"28 1","pages":"132-150"},"PeriodicalIF":10.8000,"publicationDate":"2021-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"26","resultStr":"{\"title\":\"Exactness of quadrature formulas\",\"authors\":\"L. Trefethen\",\"doi\":\"10.1137/20m1389522\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases: Newton–Cotes, Clenshaw–Curtis, Gauss–Legendre, and Gauss–Hermite quadrature. Three further examples are mentioned more briefly.\",\"PeriodicalId\":49525,\"journal\":{\"name\":\"SIAM Review\",\"volume\":\"28 1\",\"pages\":\"132-150\"},\"PeriodicalIF\":10.8000,\"publicationDate\":\"2021-01-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"26\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM Review\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/20m1389522\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Review","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/20m1389522","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases: Newton–Cotes, Clenshaw–Curtis, Gauss–Legendre, and Gauss–Hermite quadrature. Three further examples are mentioned more briefly.
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Survey and Review feature papers that provide an integrative and current viewpoint on important topics in applied or computational mathematics and scientific computing. These papers aim to offer a comprehensive perspective on the subject matter.
Research Spotlights publish concise research papers in applied and computational mathematics that are of interest to a wide range of readers in SIAM Review. The papers in this section present innovative ideas that are clearly explained and motivated. They stand out from regular publications in specific SIAM journals due to their accessibility and potential for widespread and long-lasting influence.
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