Weighted nonstandard quadrature formulas based on values of linear differential operators
The paper starts with a literature survey of several extensions and variations of Gaussian quadrature formulas that make use of values for the integrand and its (higher order) derivatives in the free and/or prefixed quadrature nodes and still keep a maximal polynomial degree of exactness. The main contribution is the construction of a quadrature formula with real nodes of the form \(\int_a^b f(x)w(x)dx\approx\sum_{k=1}^n w_k \mathcal{L}[f](x_k)\) where \(\mathcal{L}[f](x)=A(x)f(x)+B(x)f'(x)\) is a differential operator where \(A\) and \(B\) are predefined continuous functions and \(B(x)>0\) in \([a,b]\). The integrand is continuously differentiable and vanishes at \(\lambda\in\mathbb{R}\). Note that \(\lambda\) need not be in \([a,b]\) but continuity conditions need to hold in the interval that contains both \([a,b]\) and \(\lambda\). It is proved how the nodes and the weights can be computed such that the quadrature has algebraic degree of exactness \(2n\). It boils down to finding nodes and weights for a Gaussian quadrature formula for an auxiliary weight. The latter is characterized by its moments. Several numerical examples based on Mathematica software illustrate the result.
scientific article; zbMATH DE number 7489833
scientific article