Speeding up simple Mathematica expressions?

*To*: mathgroup at smc.vnet.net*Subject*: [mg63232] Speeding up simple Mathematica expressions?*From*: AES <siegman at stanford.edu>*Date*: Tue, 20 Dec 2005 04:19:29 -0500 (EST)*Organization*: Stanford University*Sender*: owner-wri-mathgroup at wolfram.com

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.

**Follow-Ups**:**Re: Speeding up simple Mathematica expressions?***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Speeding up simple Mathematica expressions?***From:*"Carl K. Woll" <carlw@wolfram.com>