{"title":"从不等式的角度看中点型规则","authors":"P. Cerone, S. Dragomir","doi":"10.1201/9780429123610-4","DOIUrl":null,"url":null,"abstract":"The article investigates interior point rules which contain the midpoint as a special case, and obtains explicit bounds through the use of a Peano kernel approach and the modern theory of inequalities. Thus the simplest open Newton-Cotes rules are examined. Both Riemann-Stieltjes and Riemann integrals are evaluated with a variety of assumptions about the integrand enabling the characterisation of the bound in terms of a variety of norms. Perturbed quadrature rules are obtained through the use of Gruss, Chebychev and Lupas inequalities, producing a variety of tighter bounds. The implementation is demonstrated through the investigation of a variety of composite rules based on inequalities developed. The analysis allows the determination of the partition required that would assure that the accuracy the result would be within a prescribed error tolerance. It is demonstrated that the bounds of the approximations are equivalent to those obtained from a Peano kernel that produces Trapezoidal type rules.","PeriodicalId":139586,"journal":{"name":"Handbook of Analytic-Computational Methods in Applied Mathematics","volume":"181 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"97","resultStr":"{\"title\":\"Midpoint-Type Rules from an Inequalities Point of View\",\"authors\":\"P. Cerone, S. Dragomir\",\"doi\":\"10.1201/9780429123610-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The article investigates interior point rules which contain the midpoint as a special case, and obtains explicit bounds through the use of a Peano kernel approach and the modern theory of inequalities. Thus the simplest open Newton-Cotes rules are examined. Both Riemann-Stieltjes and Riemann integrals are evaluated with a variety of assumptions about the integrand enabling the characterisation of the bound in terms of a variety of norms. Perturbed quadrature rules are obtained through the use of Gruss, Chebychev and Lupas inequalities, producing a variety of tighter bounds. The implementation is demonstrated through the investigation of a variety of composite rules based on inequalities developed. The analysis allows the determination of the partition required that would assure that the accuracy the result would be within a prescribed error tolerance. It is demonstrated that the bounds of the approximations are equivalent to those obtained from a Peano kernel that produces Trapezoidal type rules.\",\"PeriodicalId\":139586,\"journal\":{\"name\":\"Handbook of Analytic-Computational Methods in Applied Mathematics\",\"volume\":\"181 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-06-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"97\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Handbook of Analytic-Computational Methods in Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1201/9780429123610-4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Handbook of Analytic-Computational Methods in Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1201/9780429123610-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Midpoint-Type Rules from an Inequalities Point of View
The article investigates interior point rules which contain the midpoint as a special case, and obtains explicit bounds through the use of a Peano kernel approach and the modern theory of inequalities. Thus the simplest open Newton-Cotes rules are examined. Both Riemann-Stieltjes and Riemann integrals are evaluated with a variety of assumptions about the integrand enabling the characterisation of the bound in terms of a variety of norms. Perturbed quadrature rules are obtained through the use of Gruss, Chebychev and Lupas inequalities, producing a variety of tighter bounds. The implementation is demonstrated through the investigation of a variety of composite rules based on inequalities developed. The analysis allows the determination of the partition required that would assure that the accuracy the result would be within a prescribed error tolerance. It is demonstrated that the bounds of the approximations are equivalent to those obtained from a Peano kernel that produces Trapezoidal type rules.