Re: Speeding up simple Mathematica expressions?

*To*: mathgroup at smc.vnet.net*Subject*: [mg63247] Re: Speeding up simple Mathematica expressions?*From*: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>*Date*: Tue, 20 Dec 2005 23:35:32 -0500 (EST)*Organization*: The Open University, Milton Keynes, U.K.*References*: <do8ioc$rvd$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

"AES" <siegman at stanford.edu> a écrit dans le message de news: do8ioc$rvd$1 at smc.vnet.net... | 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. | Hi, As a starter, you should use *NSum* rather than *Sum* (I have got an increase by a factor three just by doing that): In[1]:= f[a_, x_] := Sum[Exp[-(Pi a)^2 n^2 - ((x - n Sqrt[1 - (Pi^2 a^4)])/ a)^2], {n, -Infinity, Infinity}] In[3]:= Plot[f[0.1, x], {x, 0, 3}] // Timing Out[3]= {642.359 Second, \[SkeletonIndicator]Graphics\[SkeletonIndicator]} In[4]:= g[a_, x_] := NSum[Exp[-(Pi a)^2 n^2 - ((x - n Sqrt[1 - (Pi^2 a^4)])/ a)^2], {n, -Infinity, Infinity}] In[5]:= Plot[g[0.1, x], {x, 0, 3}] // Timing Out[5]= {181.031 Second, \[SkeletonIndicator Graphics\[SkeletonIndicator]} Best regards, /J.M.