I would like to numerically integrate a smooth function (to be precise, I am computing differential entropies of Gaussian mixtures of the form $\int f(x) \log(1/f(x)) dx$).
Is there any software package or library that gives me the exact answer up to X digits (and potentially more), where X can be rather large? The GSL numerical integration seems to provide a solution with its claim:
The algorithms attempt to estimate the absolute error ABSERR = |RESULT - I| in such a way that the following inequality holds,
|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
but first I don't understand what is exactly meant by "attempt" (boldface added above) and second I would like to go beyond machine floating point if possible.
I am OK if the software gives up (if the requested accuracy/precision is too severe), but once an answer is out, it ought to be rather reliable. (Otherwise, what would be the point of computing it to begin with?)
This must be a very basic problem in numerical integration, but my search over the past few weeks have been futile. Any thread would be greatly appreciated.