Try to use Sobol sequences. More efficient (with respect to speed and precision) than Monte Carlo methods - my modest view... You'll find something here for a start

http://en.wikipedia.org/wiki/Sobol_sequence

Dear Markus,

Thanks for your info. For sure I will look at it.

Thanks once again.

George

An important method called Clenshaw–Curtis quadrature is important for accuracy , and linked to Chebychev polynomials, I suggest that you look here http://www.arc.vt.edu/about/personnel/krometis/krometis_cubature_slides_2013.pdf

Hi,

how many dimensions you are interested in? I am pretty sure, that there are methods that work well for cubature problems in few dimensions but cease to be effective in high dimensional problems.

Best regards

Herbert

If you're just integrating some functions in 2D or 3D, it's pretty simple. Here's a link where you can download 1D, 2D, and 3D Gauss Quadrature functions in C as well as other related stuff: http://dudleybenton.altervista.org/index.html If it's more complicated, such as experimental or computational data, I'd recommend orthogonal polynomials. The correct ones depend on your boundary conditions. See the section in Abramowitz & Stegun at http://people.math.sfu.ca/~cbm/aands/ If it's sparse data within a volume, use the 3D or 4D interpolation feature in TP2 at the first link and then integrate that.

Dear All,

Thank you very much for your suggestions.

I have a 5 and 7 dimensional integrtand functions with sharp peak.

Monte-Carlo method works very nice but I got problems with random number generators.

thank you all once again.

george

What problems are you having with random number generators and how does this impact the integration process? If you know you have a sharp peak, then brute force sampling in 5 to 7 dimensions would take a prohibitive amount of run time. I presume you've tried subdividing the domain? How about a transformation of variables that would stretch out the region near the peak(s) and contract everywhere else so as to concentrate your sampling. This, for instance, is how you use Gauss Quadrature to integrate from zero to infinity or conversely to integrate over a singularity, as GQ doesn't use the end points, so you don't have to actually evaluate the function at infinity or the singular points.

I made, long ago, a simple procedure in Fortran 77 which I attach. I do not have

a description now, but you may understand what it does: It assumes a brick-shaped

integration region, but you may multiply the integrand by a function which is 0

outside of the integration range. The function works be subdividing the space into

'bricks' of dimension n and using the average of the vertex values of the integrand.

The accuracy is prescribed. The attached version is applicable for n

Dear George, 

The search for a good method of integration makes sense in the presence of a priori information about the functions. Do you have this kind of information?

Is your function a known function:  "elementary," "transcendental," or  experimental?  If so, can you tell us the function?

Dear All,

Thanks a lot for your answers. I returned to this problem now and I will look very carefully all your suggestions and very soon may be will require more help :)

Happy New Year!

George

Sorry to be SO long in responding, but I've been really busy and I haven't been on RG in quite a while. Here's a link to every method for numerical integration as well as 2D and 3D. I think there are also two different ways to integrate from zero to infinity in there too. There's a lot more stuff at the second link.

http://dudleybenton.altervista.org/software/NumericalIntegration.zip

http://dudleybenton.altervista.org/software/index.html

If you need to integrate over 10, 20, 30 dimensions, ellipsoidal nested sampling is recommended.

Bye,

Carlo