Evaluate integral on many points (Cubature.jl ?)

[Tamas_Papp]

@dlfivefifty

Not necessarily for non-polynomial functions:

The article you link compares Clenshaw-Curtis to GQ, while I was comparing GQ to the naive approach of “simply evaluate the function a hundred or so times along an interval, and then fit a high order polynomial via regression”. The article you link is super-interesting, but I still think that GQ dominates the solution suggested (even though Clenshaw-Curtis may be better).

[dlfivefifty]

@Tamas_Papp

he article you link is super-interesting, but I still think that GQ dominates the solution suggested (even though Clenshaw-Curtis may be better).

The article just says the errors are comparable. Since FastGaussQuadrature.jl is O(n) while Clenshaw-Curtis is O(n*log n), which one is better depends on n.

(ApproxFun is essentially doing adaptive Clenshaw-Curtis when you use it for integrals. Note that when you have a Fun The transform step has been done so it goes down to O(n).)

[miguelraz]

Thank you very much! Will try to read.

[MikaelSlevinsky]

@stevengj

It sounds like the exponential length increase comes because you are including the same expression more than once, which also hurts performance (unless the compiler can eliminate the common subexpressions). One thing you could do is to assign each expression to a temporary variable before using it. Alternatively, just overwrite the variables exactly as you would in a loop (but without the runtime loop, just an unrolled sequence of statements).

I realize I’m a little late to the conversation, but I just thought I’d add that I was aware that the @clenshaw macro in ApproxFun grew the intermediate expressions exponentially (before the compiler untangles it and produces nice O(n) machine code). I posted in the old julia-users group to seek help as it was just a personal experiment with meta-programming.

If a good solution is available, a PR in ApproxFun would be appreciated!

[piever]

Sure, I believe the code snippet I posted above works and is faster than a simple function, I’ll test a bit more and make a PR.

EDIT: here it is.