Genz's Methods

These methods all use a transformation of the original integration region, to the unit hypercube [0,1]^m.  Beginning with ${\bf x}= C{\bf y}$, $P({\bf b})$ becomes

$$P({\bf b}) = (2\pi)^{-\frac{m}{2}} \int_{-\infty}^{b'_1({\bf... ... \int_{-\infty}^{b'_m({\bf y})} e^{-\frac{y_m^2}{2}} d{\bf y},$$

with $b'_i({\bf y}) = (b_i - \sum_{j=1}^{i-1}c_{i,j}y_j)/c_{i,i}$.

Next, let $y_i = \Phi^{-1}(z_i)$, for i = 1,2,..m, where
$\Phi(y_i) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_i} e^{-\frac{t^2}{2}}dt$, so that $\frac{1}{\sqrt{2\pi}}e^{-\frac{y_i^2}{2}}dy_i = dz_i$ and

$$P({\bf b}) = \int_0^{e_1} \int_0^{e_2(z_1)} ... \int_0^{e_m(z_1 , z_2 , ... , z_{m-1})} d{\bf z},$$

with $e_i(z_1, ..., z_{i-1}) = \Phi((b_i - \sum_{j=1}^{i-1}c_{i,j}\Phi^{-1}(z_j))/c_{i,i})$.

Finally, let z_i = e_iw_i, for i = 1,...,m, so dz_i = e_idw_i, and

$$P({\bf b}) = e_1\int_0^1e_2({\bf w}) ... \int_0^1e_m({\bf w}) \int_0^1 d{\bf w},$$

with $e_i({\bf w}) = \Phi((b_i-\sum_{j=1}^{i-1}c_{i,j}\Phi^{-1}(e_j({\bf w})w_j))/c_{i,i})$.

The innermost integral has value equal to one, so the number of integration variables can be reduced to m-1, and standard multidimensional numerical integration methods can be used for the transformed

$$P({\bf b}) = \int_0^1 \int_0^1...\int_0^1 f(w_1, w_2, ..., w_{m-1})d{\bf w},$$

with $f({\bf w}) = e_1e_2({\bf w})...e_{m}({\bf w}).$

Genz (1992) originally considered using both a crude Monte-Carlo method and a subregion adaptive method (Berntsen, Espelid and Genz, 1990) to integrate $f({\bf w})$. More recently, Beckers and Haegemans (1992) successfully used lattice rules for the integration of f(w).