Re: Speeding up simple Mathematica expressions?

*To*: mathgroup at smc.vnet.net*Subject*: [mg63250] Re: [mg63232] Speeding up simple Mathematica expressions?*From*: "Carl K. Woll" <carlw at wolfram.com>*Date*: Tue, 20 Dec 2005 23:35:35 -0500 (EST)*References*: <200512200919.EAA28491@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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}]; > Have you considered using NSum instead of Sum? f[a_?NumericQ,x_?NumericQ]:=NSum[ etc. ] works considerably faster. Carl Woll Wolfram Research > (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.

**References**:**Speeding up simple Mathematica expressions?***From:*AES <siegman@stanford.edu>