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

<br /> \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 <br />

 

[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&lt;=w_k&lt;=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.