Error estimates of Gauss-type quadrature rules for variable-sign weight functions

Abstract

The n-point Gauss quadrature formula Gn is known to be a unique optimal interpolatory quadrature rule. In practical applications of Gn, it is important to be able to estimate its error. For that purpose, (2n+1)-point extensions that inherit the n nodes of Gn, such as the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules, can be used. When Gn is applied, the weight function (or the measure) is usually assumed to be non-negative on the integration interval. In the present paper, we consider the recently introduced n-point Gauss-type quadrature formula Qn with respect to weight functions that change the sign in the interior of the integration interval. To estimate the error of Qn, we construct its (2n+1)-point extensions that inherit the n nodes of Qn and that are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Numerical examples illustrate the accuracy of the error estimates obtained by these extensions.