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.