Stratified cubature rules are proposed to approximate double integrals defined on the real positive semiaxis. In particular, anti-Gauss cubature formulae are introduced and averaged cubature schemes are developed. Some of their appropriate modifications are also studied. Several numerical experiments are given to testify the performance of all the formulae.

Consider the problem of approximating an integral of a real-valued integrand on a real interval by a Gauss quadrature rule. The classical approach to estimate the quadrature error of a Gauss rule is to evaluate an associated Gauss–Kronrod rule and compute the difference between the value of the Gauss–Kronrod rule and that of the Gauss rule. However, for a variety of measures and a number of nodes of interest, Gauss–Kronrod rules do not have real nodes or positive weights. This makes these rules impossible to apply when the integrand is defined on a real interval only. This has spurred the development of several averaged Gauss quadrature rules for estimating the quadrature error of Gauss rules. A significant advantage of the averaged Gauss rules is that they have real nodes and positive weights also in situations when Gauss–Kronrod rules do not. The most popular averaged rules include Laurie’s averaged Gauss quadrature rules, optimal averaged Gauss quadrature rules, weighted averaged Gauss quadrature rules, and two-measure-based generalized Gauss quadrature rules. This paper reviews the accuracy, numerical construction, and internality of averaged Gauss rules.