RE: Re: fourier transform time

*To*: mathgroup at smc.vnet.net*Subject*: [mg36464] RE: [mg36446] Re: fourier transform time*From*: "DrBob" <drbob at bigfoot.com>*Date*: Sun, 8 Sep 2002 03:30:40 -0400 (EDT)*Reply-to*: <drbob at bigfoot.com>*Sender*: owner-wri-mathgroup at wolfram.com

Caution: the definition b[n_] := Evaluate[(2/L)*Integrate[f[x]*Sin[n*Pi*x/L], {x, 0, L}]]; doesn't immediately compute the integral unless f is already defined at this point, and in that case you may as well write b[n_] = (2/L)*Integrate[f[x]*Sin[n*Pi*x/L], {x, 0, L}]; instead. If b will be computed more than once for the same n, its even better to do it THIS way (if f and L do not change): f[x_] = Cos[x] (* for example *) Simplify[Integrate[f[x]*Sin[n*Pi*(x/L)], {x, 0, L}]] (L*((-n)*Pi + n*Pi*Cos[L]*Cos[n*Pi] + L*Sin[L]*Sin[n*Pi]))/ ((L - n*Pi)*(L + n*Pi)) b[n_] := b[n] = (L*((-n)*Pi + n*Pi*Cos[L]*Cos[n*Pi] + L*Sin[L]*Sin[n*Pi]))/ ((L - n*Pi)*(L + n*Pi)) Bobby -----Original Message----- From: Tom Burton [mailto:tburton at brahea.com] To: mathgroup at smc.vnet.net Subject: [mg36464] [mg36446] Re: fourier transform time On 9/6/02 12:23 AM, in article al9ldd$b2j$1 at smc.vnet.net, "Steve Story" <sbstory at unity.ncsu.edu> wrote: > I don't use Mathematica > because of the speed, but should it be this slow? Mathematica has become more competitive in the speed department in recent years. See for example the attached comparison (not sent to newsgroup) by Stephan Steinhaus (steinhaus-net.de). So when Mathematica takes a very long time, you should investigate. In this case inserting Evaluate[] in two places In[91]:=b[n_] := Evaluate[(2/L)*Integrate[f[x]*Sin[n*Pi*x/L], {x, 0, L}]]; .... In[104]:=Timing[Plot[Evaluate[FS[120, x]], {x, 0, 2}]] Out[104]={0.18 Second,\[SkeletonIndicator]Graphics\[SkeletonIndicator]} speeds the process enormously (18 milliseconds to plot 120 terms on my feeble old 500MHz PowerBook). Why was it so slow before? When I switch from an ordinary numerical language to Mathematica, I enter into an implicit bargain with Mathematica: the software will go the extra mile to get me a good answer, including (1) using exra precision (sometimes without being asked) and (2) carrying around unevaluated mathematical expressions (usually without being asked) that could possibly be evaluated more appropriately at a later time. Most tools cannot do either of these things, so I don't have to worry about it, except for the bad answers that result now and then. But I need to take care that Mathematica does not burden itself unnecessarily. That's my side of the bargain. Number (2) is the issue here. Your definition of b[n] is written so that Mathematica analytically evaluates b separately for each n. But you know in this case that the integration can be done safely once for all n. So do it! The huge difference, though, comes from pre-evaluating the argument to Plot. Read the on-line help! You should pre-evaluate where possible. In some cases, the most common of which involve branching within the definition of function to plot, you cannot pre-evaluate so, in keeping with the bargain, Mathematica goes the extra mile and holds back just in case. You need to steer it into the shortcut when it's OK. Hope this helps, Tom Burton --