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I have been running into the following integral again and again:

Let $S^{n-1}= \{x \in \mathbb{R}^{n} \: | \: ||x||=1 \}$ and let $\lambda_{S^{n-1}}$ denote the surface measure over $S^{n-1}$ as defined in Stroock (2000) page 86.

Consider a fixed symmetric, positive definite matrix $Q$ of dimension $n \times n$, and a fixed scalar $a\in \mathbb{R}_{+}$

Question 1) Do you know if there is a closed form solution for the integral:

$$\int_{S^{n-1}} \exp\Big(a \omega'Q\omega \Big) \lambda_{S^{n-1}} (d \omega) $$

When $n=2$, I can express this integral as a modified Bessel function of the first kind $I_{v}(x)$ with $v=0$ evaluated at the eigenvalues of $Q$.

Question 2) Any suggestion about good numerical method for solving this integral?

Thanks!

*Stroock (2000) "A concise introduction to the theory of integration"

  • 1
    Just a quick, half baked idea: Due to rotational symmetry, you may as well assume that $Q$ is diagonal. Now it seems to me that you can do one dimension at a time, getting a single integral with an integrand involving the answer for $S^{n-2}$. Iterate.2012-08-03
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    what is $\omega'$?2012-08-03
  • 0
    I guess $\omega'$ means $\omega^t$, the transposed vector of $\omega$.2012-08-03
  • 0
    Yes, sorry: $\omega'$ is the transpose of $\omega$.2012-08-03
  • 0
    Thanks to HHO for the half baked idea. Let me work that out.2012-08-03
  • 0
    Got something from the answer below?2012-08-11

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