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Re: Speeding up simple Mathematica expressions?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg63248] Re: [mg63232] Speeding up simple Mathematica expressions?
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Tue, 20 Dec 2005 23:35:33 -0500 (EST)
*References*: <200512200919.EAA28491@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On 20 Dec 2005, at 18:19, AES wrote:
> I'd appreciate some practical advice on speeding up some simple
> function
> evaluations.
>
> I'm evaluating a series of functions of which a typical example is
>
> f[a_, x_] := Sum[
> Exp[-(Pi a)^2 n^2 -
> ((x - n Sqrt[1 - (Pi^2 a^4)])/a)^2],
> {n, -Infinity, Infinity}];
>
> (The function is essentially a set of narrow gaussian peaks located
> at x
> ? n Sqrt[1 - (Pi a^2)^2] ? n , with the peak amplitudes dropping off
> rapidly with increasing x.)
>
> Despite being a fairly simple function, this evaluates very slowly
> on my
> iBook G4 -- takes a long time to make a plot of say f[0.1, x] for
> 0 <
> x < 3. What can or should I do to speed this up?
>
> a) If this were back in early FORTRAN days, I'd surely pull the
> square
> root outside the sum -- do something like
>
> f[a_, x_] := Module[{b},
> b=Sqrt[1 - (Pi a^2)^2];
> Sum[Exp[-(Pi a n)^2 - ((x - n b)/a)^2];
>
> Is Mathematica smart enough to do that automatically, without the
> Module[] coding? Is the added overhead of the Module[] small enough
> that it's worthwhile for me to do it? Is there some other way to
> "compile" the function for a given value of a?
>
> b) Since I mostly want just plots of the first two or three peaks,
> and
> 1% accuracy should be fine, maybe I can cut the accuracy options in
> Plot[ ]. If so, how best to do this? (I've not played with those
> somewhat confusing options before.)
>
> c) Since the individual peaks have very little overlap for a < 0.2,
> maybe I can truncate the series to a small range of n?
>
> Obviously I can experiment with these and other approaches, but it's
> tedious. If any gurus have suggestions on a good quick approach, I'll
> be glad to hear them.
>
There is one very obvious thing you could try. Use NSum instead of
Sum. Just compare these two timings:
f[a_, x_] := Sum[
Exp[-(Pi a)^2 n^2 -
((x - n Sqrt[1 - (Pi^2 a^4)])/a)^2],
{n, -Infinity, Infinity}];
g[a_, x_] := NSum[
Exp[-(Pi a)^2 n^2 -
((x - n Sqrt[1 - (Pi^2 a^4)])/a)^2],
{n, -Infinity, Infinity}];
f[0.2,0.3]//N//Timing
{0.396954 Second,0.105403}
g[0.2,0.3]//Timing
{0.009731 Second,0.105403}
Looking at the difference in timings and the accuracy of the results
I do not see much need for any further ideas, do you?
Andrzej Kozlowski
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