Evaluate multivariate integral over a simplex - help

[winterfors]

I would need to evaluate the integral

$$\int\limits_{x\in S}[x+x_A]^TA[x+x_A]\exp\left(-\frac{1}{4}[x+x_B]^TB[x+x_B]\right)dx$$

where
$x$ is a column n-vector
$x_A$ and $x_B$ are column n-vector constants
$A$ and $B$ are $n\times n$ (symmetrical) matrices

taken over the volume $S$, which is a n-simplex* in $\Re^n$.

*A simplex is the convex hull of (volume between) n+1 points in $\Re^n$. In $\Re^2$ a simplex is a triangle, in $\Re^3$ a tetrahedron, etc. See http://en.wikipedia.org/wiki/SimplexI have tried to do it analytically, but the interdependence of the integration limits in each dimension (since it's over a simplex) make Matematica or Maple fail to evaluate even simple cases such as for n=2.

Approximating the integrand as a multivariate Taylor series makes it easier to integrate, but does not approximate the integrand very well if you truncate the series at a reasonable degree (the number of terms grows exponentially with the degree for a multivariate Taylor expansion)

Monte Carlo approximation isn't really an option, since I need high numerical efficiency in the evaluation, so I'm really at a loss what to to here.Does anyone have an idea?

 

[bpet]

How did you parameterize the simplex?

 

[winterfors]

The simplex is defined by n+1 points in $\Re^n$. It can be turned into the volume "under" a standard (n-1)-simplex by a linear coordinate transformation, yielding something like

$$\int\limits_{x_1=0}^{1}<br /> \int\limits_{x_2=0}^{1-x_1} ...<br /> \int\limits_{x_n=0}^{1-\sum\limits_{i=1}^{n-1}x_i}<br /> [x+x_A]^TA[x+x_A]\exp\left(-\frac{1}{4}[x+x_B]^TB[x+x_B]\right)<br /> dx_n ... dx_2 dx_1$$

 

[bpet]

More generally the points can be defined by $$x=w_1x_1+\ldots+w_{n+1}x_{n+1}$$ where $$w_1+\ldots+w_{n+1}=1$$ and $$0<=w_k<=1$$, and this results in a similar integral to the one you wrote.

But there might not be a closed-form solution to more than 1 of the n required integrals anyway.