1Blurb::
2Cubature using Stroud rules and their extensions
3Description::
4Multi-dimensional integration by Stroud cubature rules
5\cite stroud and extensions
6\cite xiu_cubature, as specified with \c cubature_integrand.
7   A total-order
8   expansion is used, where the isotropic order \e p of the
9   expansion is half of the integrand order, rounded down. The
10   total number of terms \e N for an isotropic total-order expansion
11   of order \e p over \e n variables is given by
12   \f[N~=~1 + P ~=~1 + \sum_{s=1}^{p} {\frac{1}{s!}}
13   \prod_{r=0}^{s-1} (n + r) ~=~\frac{(n+p)!}{n!p!}\f]
14   Since the maximum integrand order is currently five for normal
15   and uniform and two for all other types, at most second- and
16   first-order expansions, respectively, will be used. As a result,
17   cubature is primarily useful for global sensitivity analysis,
18   where the Sobol' indices will provide main effects and, at most,
19   two-way interactions. In addition, the random variable set must
20   be independent and identically distributed (\e iid), so the use
21   of \c askey or \c wiener transformations may be required to
22   create \e iid variable sets in the transformed space (as well as
23   to allow usage of the higher order cubature rules for normal and
24   uniform). Note that global sensitivity analysis often assumes
25   uniform bounded regions, rather than precise probability
26   distributions, so the \e iid restriction would not be problematic
27   in that case.
28
29Topics::
30Examples::
31Theory::
32Faq::
33See_Also::
34