Blurb::
Cubature using Stroud rules and their extensions
Description::
Multi-dimensional integration by Stroud cubature rules 
\cite stroud and extensions 
\cite xiu_cubature, as specified with \c cubature_integrand. 
   A total-order
   expansion is used, where the isotropic order \e p of the
   expansion is half of the integrand order, rounded down. The
   total number of terms \e N for an isotropic total-order expansion
   of order \e p over \e n variables is given by
   \f[N~=~1 + P ~=~1 + \sum_{s=1}^{p} {\frac{1}{s!}} 
   \prod_{r=0}^{s-1} (n + r) ~=~\frac{(n+p)!}{n!p!}\f]
   Since the maximum integrand order is currently five for normal
   and uniform and two for all other types, at most second- and
   first-order expansions, respectively, will be used. As a result,
   cubature is primarily useful for global sensitivity analysis,
   where the Sobol' indices will provide main effects and, at most,
   two-way interactions. In addition, the random variable set must
   be independent and identically distributed (\e iid), so the use
   of \c askey or \c wiener transformations may be required to
   create \e iid variable sets in the transformed space (as well as
   to allow usage of the higher order cubature rules for normal and
   uniform). Note that global sensitivity analysis often assumes
   uniform bounded regions, rather than precise probability
   distributions, so the \e iid restriction would not be problematic
   in that case.

Topics::
Examples::
Theory::
Faq::
See_Also::	
