I'd like to know about the best numerical methods for approximating an integral. Unfortunately, I want to know about a fairly general case, so I cannot give a lot of information.
Essentially, I have an integral that is full of exponential functions and little else. I therefore assume as given, although possibly incorrectly, that the power series expansion of the integrand can be integrated to give an exact value to the integral. In other words, we can take the power series expansion of the integrand (the function inside the integral), and integrate it to obtain the same value as integrating the integral itself gives.
I then "define" the value $n$, saying that we need at least $n$ terms of the power series expansion to integrate to the wanted degree of accuracy/precision. I also "define" $m$, such that by numerical methods, the integral must be evaluated at at least $m$ points to obtain the same degree of accuracy/precision.
Hoping I've stated the integral and defined some restraints well, I have some curiosities. I'm wondering how well numerical methods can hope to achieve these requirements. In other words, how many calculations, and how much memory, etc. is needed to approximate this integral? For example, if we know that we have to evaluate the integral at $m$ points, does it take $O(m)$ calculations to evaluate this integral, or perhaps $O(m \log(m))$? Similarly, if we know that the power series expansion requires $n$ terms to get the job done, how much computing power do we need? $O(n)$? $O(n \log(n))$? Some other value? I know that this is an extremely general case, but I'm curious as to how much computing requirements various methods take?
I'm asking because I want to compare a method that I'm experimenting with. I doubt that I can do as well as the best methods, but I would like to know what the best method is so I can eventually use it.