Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa
type: Item
instance of: scholarly article
title: Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa
zbMATH Open document ID: 1312.65032
author: Bernhard Beckermann
author: Reinaldo Martínez-Cruz
author: Jorge Bustamante
author: José M. Quesada
published in: Calcolo
publication date: 2015-03-23T00:00:00Z
review text: Let \(\sigma\) be a positive measure on a compact interval \([a,\,b]\). A positive \(n\)-point quadrature formula has the form \[ \sum_{j=1}^n \lambda_{n,j}\,f(x_{n,j}) \approx \int_a^b f(x) \,d\sigma(x) \] with distinct abscissas \(x_{n,j}\in [a,\,b]\) and positive weights \(\lambda_{n,j}\). As known, there exist \(n\)-point Gauss rules with degree of precision \(2n-1\). Further, there exist \(n\)-point (left resp. right) Radau rules with abscissa \(a\) resp. \(b\) and with degree of precision \(2n-2\), and an \(n\)-point Lobatto rule with abscissas \(a\), \(b\) and with degree of precision \(2n-3\). In this paper, the authors show that for a given \(\theta\in (a,\,b)\), there exist an \(n\)-point quasi Gauss rule, \(n\)-point quasi (left/right) Radau rule, resp. \(n\)-point quasi Lobatto rule with the additional abscissa \(\theta\) and that this quadrature formula has the degree of precision \(2n-2\), \(2n-3\), resp. \(2n-4\). A central role in this study is played by quasi-orthogonal polynomials.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 6418715
zbMATH Keywords: positive quadrature formula
zbMATH Keywords: prescribed abscissa
zbMATH Keywords: Gauss rule
zbMATH Keywords: Radau rule
zbMATH Keywords: Lobatto rule
zbMATH Keywords: degree of precision
zbMATH Keywords: quasi-orthogonal polynomials
MaRDI profile type: Publication
full work available at URL: s10092-013-0087-3
OpenAlex ID: W2151818338
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DOI: 10.1007/S10092-013-0087-3
description: scientific article; zbMATH DE number 6418715