Timeline for Rapid evaluation of multivariate normal integral

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Apr 4, 2013 at 23:13	comment	added	Brendan McKay		G-H integration with $n$ abscissae gives the exact integral of $e^{-x^2}p(x)$ when $p(x)$ is any polynomial of degree at most $2n-1$. So it gives a fair approximation when $p(x)$ can be fairly approximated by a polynomial of degree $2n-1$ over the region that matters, and $2n-1$ is a pretty high degree polynomial to play with. It will depend on the constants $a_i,b_i$. If they are all equal and $N$ is very large, the product approaches a step function and your integral is just a chunk of the guassian integral; G-H won't work in that case.
Apr 4, 2013 at 14:29	comment	added	melchimm		I should have mentioned this in my original question, but I am currently using a different quadrature rule that is inspired by Monte Carlo methods and that I believe (intuitively, I have no proof) should be more efficient than G-H. What I'm currently doing is choosing the abscissae to be centered on each of k quantile intervals for the normal pdf and giving the corresponding function evaluations equal weight in the resulting sum. Given that, as John points out below, my integrand is not well approximated by a polynomial, is there any reason to think that G-H would be more efficient?
Apr 4, 2013 at 12:08	history	answered	Brendan McKay	CC BY-SA 3.0