# Efficient Sampling

I'm trying to sample a lot of points efficiently. I'm wondering if the following method is possible.

I sample points of a function (evaluate the function) mod $n$. I.e. I calculate f(element one), f(element 2)...f(element n). I do this for all of the points in a ring. Now I take a second ring (mod $m$, for instance) and calculate the points of a function for all elements of this ring.

So my idea is this: I've sampled only $n+m$ points, but the combinations of these sample points number $n \cdot m$. I'd like to calculate the sum of (evaluations of) this larger set of points efficiently. I'm wondering if this is possible.

I'd like to know as much about this idea as possible. I'd like to know related ideas, what's possible, or just general ideas on how to do this. Any pointers would constitute an acceptable answer to me, and an actual method would be absolutely wonderful.

I realize that this may be more of a numerical method than mathematics, but I'm hoping that everyone will accept this question, since it may allow new methods of integration as a result.

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As far as I know, numerical methods are a part of mathematics. But I'm confused by your question. What are the domain and codomain of your function $f$? – Rahul Oct 26 '10 at 22:48
I tried to simplify the question. I want to sample $f$, whose domain and codomain are both complex. However, I can represent this by $\mathbb{C} /n \mathbb{R}$ if I have my terminology correct. However, I'd be satisfied with an example that uses only rings of integers. I'd also be satisfied with an example function whose domain and codomain are also integers or naturals. – Matt Groff Oct 26 '10 at 22:57
"new methods of integration" - I presume you've looked at the literature on "adaptive quadrature"? – J. M. Oct 27 '10 at 14:07
I haven't. I just got really excited and wrote that. I'll learned a lesson from this, and hope to be more scientific in the future. Again, I'd like to know about everything related - "adaptive quadrature" articles seem promising. – Matt Groff Oct 27 '10 at 16:02