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