Gauss-Radau and Gauss-Lobatto quadratures with double end points
type: Item
instance of: scholarly article
title: Gauss-Radau and Gauss-Lobatto quadratures with double end points
zbMATH Open document ID: 0727.65012
DOI: 10.1016/0377-0427(91)90094-Z
author: Walter Gautschi
author: Ken Li
published in: Journal of Computational and Applied Mathematics
publication date: 1991-01-01T00:00:00Z
review text: In this paper Gauss-Radau and Gauss-Lobatto quadrature rules are developed having end points of multiplicity 2. These formulae seem to be useful for special integrands as well as when applied to boundary value problems. First, the Gauss-Radau formula on [-1,1] is derived with a double node at \(-1,\) \[ \int^{1}_{-1}f(t)w(t)dt=\kappa^ R_ 0f(-1)+\kappa^ R_ 1f'(-1)+\sum^{n}_{\nu =1}\lambda^ R_{\nu}f(\tau^ R_{\nu})+E^ R_ n(f); \] this is done both for general weight functions and all four of Chebyshev type \((w(t)=(1-t^ 2)^{-1/2}\), \((1-t^ 2)^{1/2}\), \((1-t)^{-1/2}(1+t)^{1/2}\), \((1- t)^{1/2}(1+t)^{-1/2})\), yielding explicit expressions for \(\kappa^ R_ i\) and \(\lambda^ R_ i.\) In a second part, the analogon is carried out for Gauss-Lobatto rules on \([-1,1]\) with double nodes at -1 and 1, \[ \int^{1}_{- 1}f(t)w(t)dt=\kappa^ L_ 0f(-1)+\kappa^ L_ 1f'(- 1)+\sum^{n}_{\nu =1}\lambda^ L_{\nu}f(\tau^ L_{\nu})+\mu^ L_ 0f(1)-\mu^ L_ 1f'(-1)+E^ L_ n(f). \] Here, the symmetry of the problem is widely exploited. Finally, numerical examples are given demonstrating advantages of the new formulae in special cases.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 4201462
zbMATH Keywords: double end points
zbMATH Keywords: explicit quadrature formula
zbMATH Keywords: Chebyshev weight functions
zbMATH Keywords: Gaussian quadrature formulae
zbMATH Keywords: Gauss-Radau formula
zbMATH Keywords: Gauss- Lobatto rules
zbMATH Keywords: numerical examples
describes a project that uses: Algorithm 236
MaRDI profile type: Publication
cites work: Q5508667
cites work: Q3938409
cites work: Q3996470
cites work: Corrections to ''On the remainder term for analytic functions of Gauss- Lobatto and Gauss-Radau quadratures''
cites work: The remainder term for analytic functions of Gauss-Radau and Gauss- Lobatto quadrature rules with multiple end points
cites work: Q3671491
cites work: On the calculation of Jacobi matrices
full work available at URL: 0377-0427(91)90094-z
OpenAlex ID: W2070572391
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description: scientific article; zbMATH DE number 4201462