Some theoretical aspects of generalised quadrature methods.
Generalized quadrature methods for the numerical approximation of integrals \[ Q(f)=\int_a^bw(x)f(x)dx \] with a given irregular oscillatory weight function \(w\), satisfying \(Lw=0\) for a differential operator \(L\), are constructed using Lagrange's identity. Forcing exactness for a set of basis functions leads to a set of linear equations for the weights \(a_k\) of the quadrature formula \(Q(f) \approx \sum a_kf(x_k)\). These equations are solved numerically. Troubles due to ill-conditioning can occur. The paper is mainly concerned with the question, of how the numerical errors in the computed weights \(a_k\) affect the accuracy of the approximation to the integral \(Q(f)\). Four examples illustrate the theoretical predictions.
scientific article; zbMATH DE number 1967208