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Evaluate multivariate integral over a simplex - help!

  1. Mar 6, 2009 #1
    I would need to evaluate the integral

    [tex]\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[/tex]

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

    taken over the volume [itex]S[/itex], which is a n-simplex* in [itex]\Re^n[/itex].

    *A simplex is the convex hull of (volume between) n+1 points in [itex]\Re^n[/itex]. In [itex]\Re^2[/itex] a simplex is a triangle, in [itex]\Re^3[/itex] a tetrahedron, etc. See http://en.wikipedia.org/wiki/Simplex


    I 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?
     
    Last edited: Mar 6, 2009
  2. jcsd
  3. Mar 8, 2009 #2
    How did you parameterize the simplex?
     
  4. Mar 8, 2009 #3
    The simplex is defined by n+1 points in [itex]\Re^n[/itex]. It can be turned into the volume "under" a standard (n-1)-simplex by a linear coordinate transformation, yielding something like

    [tex]
    \int\limits_{x_1=0}^{1}
    \int\limits_{x_2=0}^{1-x_1} ...
    \int\limits_{x_n=0}^{1-\sum\limits_{i=1}^{n-1}x_i}
    [x+x_A]^TA[x+x_A]\exp\left(-\frac{1}{4}[x+x_B]^TB[x+x_B]\right)
    dx_n ... dx_2 dx_1
    [/tex]
     
  5. Mar 8, 2009 #4
    More generally the points can be defined by [tex]x=w_1x_1+\ldots+w_{n+1}x_{n+1}[/tex] where [tex]w_1+\ldots+w_{n+1}=1[/tex] and [tex]0<=w_k<=1[/tex], 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.
     
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