Cubatures of precision $2k$ and $2k+1$ for hyperrectangles

Abstract:

It is well known that integration formulas of precision $2k\;(2k + 1)$ for a region in n-space which is a Cartesian product of intervals can be obtained from one-dimensional Radau (Gauss) rules. The number of function evaluations in these product cubatures is ${(k + 1)^n}$. In this paper, an algorithm is given for obtaining cubatures for hyperrectangles in n-space of precision 2k, in many instances $2k + 1$, which uses $(k + 1){(k)^{n - 1}}$ nodes. The weights and nodes of these new formulas are derived from one-dimensional generalized Radau rules.