I need to compute a set of multivariate definite integrals with infinite integration domain

$$\displaystyle \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f(x_1,x_2, \ldots , x_n)\;\;dx_1 \cdots dx_n $$

With $n$ typically taking values between 80-150.

I have not found a closed form solution. Besides the function $f$ cannot be decomposed as product of unidimensional functions in $x_i$.

Does there exist a matlab toolbox or any software package with a matlab interface to do that?

Note: I am totally newbie in numerical integration. This problem comes from a totally different field. Thanks.

Edit: after some minor manipulation the integral look like this

Let $\boldsymbol{x},\boldsymbol{b}\in\mathbb{R}^n$ and $\boldsymbol{A}\in\mathbb{R}^{n\times n}$

$$ I = \int_{-\infty}^{\infty} exp ( -\frac{1}{2} x^T A x + b^T x ) \; \prod_{i=1}^n exp(-|x_i|) \; dx = \int_{-\infty}^{\infty} exp ( -\frac{1}{2} x^T A x + b^T x - \sum_{i=1}^n |x_i| ) \; dx $$

$$I=\int_{-\infty}^{\infty} \cdots\int_{-\infty}^{\infty} exp(-\frac{1}{2}\sum_{i=1}^n \sum_{j=1}^n a_{i,j}\;x_ix_j + \sum_{i=1}^n b_ix_i-\sum_{i=1}^n |x_i|)\; dx_1 \cdots dx_n$$

Whereas I try to find an analytical solution I have been playing with numerical integration. I wrote this script to integrate a multivariate normal distribution(because is an easy function and we know the result is 1). The problem is that the computation time grows with the dimension and the script is at moment too slow for my requirements. If anyone can give me advise to speed up the code I will be grateful.

```
clc
close all
clear all
pmax=30;
t=zeros(pmax,1);
result=zeros(pmax,1);
sigma=2;
for iter=1:pmax
str = [' Iteration ', num2str(iter), ' of ',num2str(pmax)];
disp(str)
p=iter;
S=(sigma^2)*eye(p);
invS=inv(S);
detS=det(S);
%x=[x1;...; xp]
x=sym('x',[p,1]);
% x'= complex conjugate transpose
% x.'= transpose without conjugation
xT=x.';
f=exp(-0.5*xT*invS*x);
fk=f;
tstart=tic;
for k=1:p
fk=int(fk,x(k),-inf,inf);
end
t(iter)=toc(tstart);
aux=1/sqrt((2*pi)^p);
aux=aux/sqrt(detS);
aux=aux*fk;
result(iter)=aux;
end
figure
plot(result)
title('Result (Ideally always 1)')
figure
plot(t)
title('time elapsed')
```

expectation propagationalgorithms for Bayesian inference, and their efficient evaluation is one of the key ingredients. Maybe you can get some ideas from those papers. $\endgroup$ – Christian Clason Jul 27 '13 at 8:41