Integrating a multidimensional integral in scipy

Question

Motivation: I have a multidimensional integral, which for completeness I have reproduced below. It comes from the computation of the second virial coefficie

Answer 1

I'll just make a couple of general comments about how to accurately do this sort of integral, but this advice is not specific to scipy (too long for a comment, even though it is not an answer).

I don't know your use case, i.e. whether you are satisfied with a `good' answer with a few digits of accuracy which could be obtained straightforwardly using Monte Carlo as outlined in Jonathan Dursi's answer, or whether you really want to push the numerical accuracy as far as possible.

I've performed analytic, Monte Carlo and quadrature calculations of virial coefficients myself. If you want to do the integrals accurately, then there are a few things you should do:

1. Attempt to perform as many of the integrals exactly as possible; it may well be that integration in some of your coordinates is quite simple.

2. Consider transforming your variables of integration so that the integrand is as smooth as possible. (This helps for both Monte Carlo and quadrature).

3. For Monte Carlo, use importance sampling for best convergence.

4. For quadrature, with 7 integrals it may just be possible to get really fast convergence using tanh-sinh quadrature. If you can get it down to 5 integrals then you should be able to get 10's of digits of precision for your integral. I highly recommend mathtool / ARPREC for this purpose, available from David Bailey's homepage: http://www.davidhbailey.com/