= Tensor-product quadrature
{title2=$Q(f)=\sum_{i_1,\ldots,i_d}w_{i_1}\cdots w_{i_d}f(x_{i_1},\ldots,x_{i_d})$}
On a product domain, apply a one-dimensional <quadrature rule> independently in each coordinate. The resulting <quadrature nodes> form a product grid and the <quadrature weights> multiply. With $m$ subdivisions per direction in fixed dimension $d$, the evaluation count is $N=O(m^d)$. If the one-dimensional rule has error $O(m^{-p})$, uniformly for the required coordinate slices, and its weight sums are bounded, telescoping the $d$ integration operators against their quadrature operators gives total error $O(m^{-p})=O(N^{-p/d})$. Thus <composite Simpson's rule> has $p=4$ for sufficiently smooth integrands, but loses efficiency as $d$ increases.
Back to article page
