Symmetric Gauss-Lobatto and modified anti-Gauss rules
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.
scientific article; zbMATH DE number 2073449