Quadrature rules using first derivatives for oscillatory integrands
The authors consider the integral of a function \(y(x)\), \(I(y(x)) = \int_{ - 1}^1 y(x) dx\) and its approximation by a quadrature rule of the form \[ Q_N (y(x)) = \sum\limits_{k = 1}^N w_k y(x_k) + \sum\limits_{k = 1}^N \alpha _k y'(x_k), \] i.e., by a rule which uses the values of both \(y\) and its derivative at nodes of the quadrature rule. They examine the cases when the integrand is either a smooth function or an \(\omega \) dependent function of the form \[ y(x) = f_1 (x)\sin (\omega x) + f_2 (x)\cos (\omega x) \] with smoothly varying \(f_1 \) and \(f_2 \). In the latter case, the widths \(w_k \) and \(\alpha _k \) are \(\omega \) dependent. Some general properties of the widths are established and some numerical illustrations are presented.
scientific article; zbMATH DE number 1758443
scientific article