Symmetric integration rules for hypercubes. II. Rule projection and rule extension

A theory is described which facilitates the construction of highdimensional integration rules. It is found that, for large $n$, an $n$-dimensional integration rule of degree $2t + 1$ man be constructed requiring a number of function evaluations of order ${2^t}{n^t}/t!$. In an example we construct a $15$-dimensional rule of degree 9 which requires 52,701 function evaluations. The corresponding number for the product Gaussian is $3 \times {10^{10}}$.