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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.

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closed as too broad by Arkamis, O.L., Danny Cheuk, Daniel Rust, Amzoti Jul 25 '13 at 14:31

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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As I (may have) mentioned in your previous questions, there is no one general method that works well for both real and complex (sines and cosines) exponentials; a method well-suited to coping with wiggles may find itself at a loss with decaying exponentials, and vice-versa. –  J. M. Nov 23 '10 at 4:14
    
I'm actually interested primarily in complex exponentials, and all exponentials will be $e^{i\cdot n}$ with varying $n$. I don't want to be vague, but I'm interested in the method with the best potential, and someone may be able to find a way to express the integrals in different terms. –  Matt Groff Nov 23 '10 at 5:26
    
A look at the literature for oscillatory integrals should be enlightening; (you might want to look out for papers by Arieh Iserles). Finding a contour to integrate around (to avoid wiggles in your integrand) should be seriously considered as well. –  J. M. Nov 23 '10 at 6:32
    
For my particular case, I fear I may be stuck with a special circular contour. I'm really attempting to find the (coefficient of the) term in the middle of a finite (power) series, if that makes sense. So I'm using a Fourier transform (I believe-I'm not well educated) and I don't know another way to single out the coefficient. I wish I could find another contour that works! –  Matt Groff Nov 23 '10 at 12:55
    
This is too old to migrate, but you might want to delete and repost on Computational Science –  robjohn Jan 23 '13 at 1:04