I need to compute many different integrals of the same function fn, essentially like this:
Xint(i) = integral(fn,0,X(i));
The problem is that X is large and computation of fn is slow, so this simple loop takes quite a long time. My question is, what is the best way to speed up this loop?
It does seem like considerable speed-up should be possible, because each time through the loop the integral computation evaluates fn at many almost-the-same points from 0 to X(i)--the range of the X(i)'s is about 400-2000. But what's the best approach? E.g. maybe just order the X's and compute the integrals in pieces, 0 to min(X), then min(X) to 2nd-smallest-X, etc (but isn't there a danger that the errors of approximation will accumulate)? Or maybe precompute fn at a bunch of X's and use trapz (but how to choose those to-be-precomputed X's to ensure the desired level of accuracy)? Or precompute fn, spline it, and then let integral work with the spline_of_fn?
Given all the work that has been done on numerical methods of integration, it seems like someone must have studied how to do this most effectively, but I can't find anything on it. So, I'd appreciate any pointers or tips.