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Re: Pure functions vs rule based functions inside NIntegrate

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  • Subject: [mg132265] Re: Pure functions vs rule based functions inside NIntegrate
  • From: Richard Fateman <fateman at>
  • Date: Tue, 28 Jan 2014 06:13:22 -0500 (EST)
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On 1/26/2014 12:05 AM, Yi Wang wrote:
> Hello,
> As I was told, only pure functions can be compiled. Also, NIntegrate
> or NDSolve first compiles a function before doing numerical
> operations.

It evaluates its first argument to an expression,
so the two examples you have here are equivalent.

For illustration, see how many times something is printed for...

f4[x_] := (Print[x]; Sin[x]); NIntegrate[f4[x],{x,0,100}]

f5 = (Print[#]; Sin[#]) & ; NIntegrate[f5[x],{x,0,100}]

(Just once)

  One of the difficulties in using Mathematica is that the semantics of 
the commands may be poorly documented or specified, and in fact may 
change between versions. Typical users adopt a trial and error approach
and hope that their notions of consistency are operationally correct.

Compare the above to   Plot[f5[x],{x,0,100}]

(many times, at least in version 9).

Of course if you truly need lots of fast multi-dimensional numerical 
integration, there may be better packages available to you.

  Thus, I would have expected that using pure functions in
> NIntegrate or NDSolve could be faster than rule based functions.
> However, in tests I didn't find such a difference. For example,
> f1[x_, y_] := Sin[Sin[x Sin[y]]]; AbsoluteTiming@NIntegrate[f1[x, y],
> {x, 0, 100}, {y, 0, 100}]
> and
> f2 = Sin[Sin[#1 Sin[#2]]] & AbsoluteTiming@NIntegrate[f2[x, y], {x,
> 0, 100}, {y, 0, 100}]
> takes about the same time. How can I tell if both are compiled, or
> neither, and why?
> Thanks! Yi

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