Symmetric Gauss-Lobatto and modified anti-Gauss rules
type: Item
instance of: scholarly article
title: Symmetric Gauss-Lobatto and modified anti-Gauss rules
zbMATH Open document ID: 1056.41017
published in: BIT
publication date: 2004-06-11T00:00:00Z
review text: Let \(\mu (t)\) be a nondecreasing function in the finite interval \([-a,a]\) and consider the integral \[ {\mathcal T}f =\int _{-a}^a f(t) \,d\mu (t), \] and the approximations given by the \(m\)-point Gauss quadrature rule \[ \mathcal G _mf =\sum _{j=1}^m f(t_j)w_j^2, \] and by the \(m+1\)-point Gauss-Lobatto quadrature rule \[ \widehat {\mathcal G} _{m+1}f =f(-a)\widehat w_1^2+\sum _{j=2}^m f(\widehat t_j)\widehat w_j^2 +f(a)\widehat w_{m+1}^2. \] The modified anti-Gauss rule considered by the authors is \[ \widetilde {\mathcal G} _{m+1,\gamma }f =\sum _{j=1}^{m+1}f(\widetilde t_j)\widetilde w_j^2, \] and requires the integration error \[ \varepsilon _{m+1,\gamma }f=(\mathcal T-\widetilde {\mathcal G}_{m+1,\gamma })f \] to satisfy \[ (\mathcal T-\widetilde {\mathcal G} _{m+1,\gamma })f=-\gamma \varepsilon _m f,\quad \forall f\in \mathbb P _{2m+1}, \] where \(\mathbb P _n\) denotes the set of polynomials of degree at most \(n\), \(\gamma \) is a constant and \(\varepsilon _mf=(\mathcal T-\mathcal G_m)f\) is the quadrature error of the Gauss rule. The main result of the paper shows that the symmetric Gauss-Lobatto rules are modified anti-Gauss rules. So, the interesting consequence that for many integrals symmetric Gauss quadrature rules and symmetric Gauss-Lobatto rules give quadrature errors of opposite sign.
reviewed by: Luigi Gatteschi
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 2073449
zbMATH Keywords: Gauss and Gauss-Lobatto quadratures
zbMATH Keywords: anti Gauss quadrature
zbMATH Keywords: error estimation
author: Daniela Calvetti
author: Lothar Reichel
MaRDI profile type: Publication
full work available at URL: b:bitn.0000007053.03860.c0
OpenAlex ID: W1996418117
DOI: 10.1023/B:BITN.0000007053.03860.C0
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description: scientific article; zbMATH DE number 2073449