A new class of quadrature rules for estimating the error in Gauss quadrature

The need to evaluate Gauss quadrature rules arises in many applications in science and engineering. It often is important to be able to estimate the quadrature error when applying an ℓ-point Gauss rule, $G_{\ell }\left(f\right)$, where f is an integrand of interest. Such an estimate often is furnished by applying another quadrature rule, $Q_k\left(f\right)$, with $k>\ell$ nodes, and using the difference $Q_k\left(f\right)-G_{\ell }\left(f\right)$ or its magnitude as an estimate for the quadrature error in $G_{\ell }\left(f\right)$ or its magnitude. The classical approach to estimate the error in $G_{\ell }\left(f\right)$ is to let $Q_k\left(f\right)$, with $k=2\ell +1$, be the Gauss-Kronrod quadrature rule associated with $G_{\ell }\left(f\right)$. However, it is well known that the Gauss-Kronrod rule associated with a Gauss rule $G_{\ell }\left(f\right)$ might not exist for certain measures that determine the Gauss rule and for certain numbers of nodes. This prompted M. M. Spalević [1] to develop generalized averaged Gauss rules, ${\hat{G}}_{2\ell +1}$, with $2\ell +1$ nodes for estimating the error in $G_{\ell }\left(f\right)$. Similarly as for $(2\ell +1)$-node Gauss-Kronrod rules, ℓ nodes of the rule ${\hat{G}}_{2\ell +1}$ agree with the nodes of $G_{\ell }$. However, generalized averaged Gauss rules are not internal for some measures. They therefore may not be applicable when the integrand only is defined on the convex hull of the support of the measure. This paper describes a new kind of quadrature rules that may be internal also when generalized averaged quadrature rules are not. The construction of the new quadrature rules is based on theory developed by Peherstorfer [2]. Their application is particularly attractive when the rule ${\hat{G}}_{2\ell +1}$ is not internal, the integrand cannot be evaluated at all its nodes, and the integrand is inexpensive to evaluate at the quadrature points. Computed examples that illustrate the performance of the new quadrature rules introduced in this paper are presented.