Gauss-type quadrature rules with respect to external zeros of the integrand

Abstract

In the present paper, we propose a Gauss-type quadrature rule into which the external zeros of the integrand (the zeros of the integrand outside the integration interval) are incorporated. This new formula with n nodes, denoted by Gn, proves to be exact for certain polynomials of degree greater than 2n−1 (while the Gauss quadrature formula with the same number of nodes is exact for all polynomials of degree less than or equal to 2n−1). It turns out that Gn has several good properties: all its nodes are pairwise distinct and belong to the interior of the integration interval, all its weights are positive, it converges, and it is applicable both when the external zeros of the integrand are known exactly and when they are known approximately. In order to economically estimate the error of Gn, we construct its extensions that inherit the n nodes of Gn and that are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Further, we show that Gn with respect to the pairwise distinct external zeros of the integrand represents a special case of the (slightly modified) Gauss quadrature formula with preassigned nodes. The accuracy of Gn and its extensions is confirmed by numerical experiments.