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I'm trying to implement a numerical integrator that should have the minimum relative error and is not slow. So I was looking for the best accepted state-of-the-art algorithm to do so but there seems to be so many approaches that I could not understand which one should I choose. So I'm turning to you for a recommendation.

Thank you for your attention,

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    I suggest you try [adaptive quadrature](http://en.wikipedia.org/wiki/Adaptive_quadrature).2012-04-17
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    There is no such thing as "one algorithm to evaluate them all". Methods for "nice integrals" are quite different from methods for, say, infinite oscillatory integrals, or Cauchy principal value integrals. That's why there are a lot of algorithms, since families of integrals might carry their own unique set of difficulties.2012-04-17
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    So your recommendation would be to have a set of solvers that would vary accordingly from some sort of integral previous analysis?2012-04-17
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    @Reonarudo : I believe in more traditional hyphenation conventions, and changed this to "best state-of-the-art numerical integral algorithm", _with_ hyphens. If it had said "What is the state of the art?", I'd have left it without hyphens. But change it back if you disagree.2012-04-17
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    Yes, and that is precisely the route taken by packages like QUADPACK...2012-04-17
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    This is a complete deviation from the original question but what kind of previous analysis should be performed then?2012-04-17
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    Offhand: check for singularities. Improper integrals often require special methods. Functions with rapidly decaying or very oscillatory factors also need special treatment. Discontinuous functions require splitting at discontinuities. Everything else is fair game for adaptive quadrature.2012-04-17
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    @J.M.but QUADPACK is one dimensional, I didn't specify in the question but my goal is something that can compute at least 3 dimension integrals. But a good example anyway. I think I'll start of from your suggestion and see where I get. Thank you.2012-04-17
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    Well, not my fault that you weren't specific at the outset, no? Efficient cubature methods remain an active area of research, and unless you have the expertise, I caution against rolling out your own implementation. You will want to look at the work already done by Ronald Cools and Terje Espelid.2012-04-17

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