Re: Mathematica 20x slower than Java at arithmetic/special functions, is

*To*: mathgroup at smc.vnet.net*Subject*: [mg115899] Re: Mathematica 20x slower than Java at arithmetic/special functions, is*From*: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>*Date*: Tue, 25 Jan 2011 04:20:30 -0500 (EST)*References*: <201101232233.RAA22629@smc.vnet.net> <ihjlu1$t63$1@smc.vnet.net>

I found a speed-up factor of about 52 using Oliver's method. This is on my quad-core laptop. So quite an improvement, but not a factor of 100. I used Evaluate[] too. I assume it works because with it the Bessel functions are evaluated at compile time (it's arguments do not depend on x and y), otherwise it's done at runtime. Cheers -- Sjoerd On Jan 24, 11:57 am, Leo Alekseyev <dnqu... at gmail.com> wrote: > Vivek, Oliver -- thanks for your input! My knowledge in using > Compile[] is somewhat lacking (mostly, due to the fact that I was > never able to get it to work well for me). In particular, I tried > using Compile[] much in the same way that Vivek has suggested, but I > neglected to use Evaluate[], which leads to a compiled function taking > substantially longer. Is there a quick explanation for why Evaluate[] > (or, in Oliver's example, a construct like > With[{code=code},Compile[{...},code]] necessary?.. > > On my (very modest) hardware, I indeed get ~25x speedup that Vivek > mentions. Oliver's code for me performs about the same (~25x > improvement) without parallelism, and 2x faster on a dual-core > machine; this actually seems reasonable since the two methods are > fairly similar. > > I should note that it seems that these optimizations are very > dependent on Mathematica 8: in particular, cfunc2 (compilation of a > compiled function evaluating over some data) in Vivek's example gives > no additional gain under Mathematica 7 (makes me curious what changed > in version 8), and RuntimeAttributes -> Listable, Parallelization -> > True options that Oliver uses are new to version 8. > > --Leo > > On Mon, Jan 24, 2011 at 4:14 AM, Vivek J. Joshi <viv... at wolfram.com> wrot= e: > > > Without going into too much detail, a simple compilation of the functio= n gives approx 6x to 25x speed up, > > > ClearAll[grid1dc]; > > grid1dc[x_,y_]=(With[{d=0.1,NN=50}, > > Sum[Re[N[d BesselJ[1,2 Pi d Sqrt[m^2+n^2]]/Sqrt[m^2+n^2+10^-7]] Exp[I 2= .0Pi (m x+n y)]],{m,-NN,NN,1},{n,-NN,NN,1}]])//N; > > > gridres1da=With[{delta=0.5,xlim=2.5,ylim=2.5}, > > Table[{x,y,grid1dc[x,y]},{x,-xlim,xlim,delta},{y,-ylim,ylim,delta}]];//= AbsoluteTiming > > {7.371354,Null} > > > Clear[cfunc]; > > cfunc = Compile[{{x,_Real},{y,_Real}},Evaluate[grid1dc[x,y]]]; > > > gridres1da2=With[{delta=0.5,xlim=2.5,ylim=2.5}, > > Table[{x,y,cfunc[x,y]},{x,-xlim,xlim,delta},{y,-ylim,ylim,delta}]];//Ab= soluteTiming > > {1.237029,Null} > > > Norm[gridres1da[[All,All,3]]-gridres1da2[[All,All,3]]]//Chop > > 0 > > > Following gives about 25x speedup, > > > Clear[cfunc2]; > > cfunc2= Compile[{{xlim,_Real},{ylim,_Real},{delta,_Real}}, > > Block[{x,y}, > > Table[{x,y,cfunc[x,y]},{x,-xlim,xlim,delta},{y,-ylim,ylim,delta}]]]; > > > gridres1da3=cfunc2[2.5,2.5,0.5];//AbsoluteTiming > > {0.292562,Null} > > > Norm[gridres1da[[All,All,3]]-gridres1da3[[All,All,3]]]//Chop > > 0 > > > Vivek J. Joshi > > Kernel Developer > > Wolfram Research Inc. > > > On Jan 24, 2011, at 4:03 AM, Leo Alekseyev wrote: > > >> I was playing around with JLink the other day, and noticed that Java > >> seems to outperform Mathematica by ~20x in an area where I'd expect > >> Mathematica to be rather well optimized -- arithmetic involving specia= l > >> functions. In my particular example, I am simply evaluating a sum o= f > >> Bessel functions. I understand that much depends on the underlying > >> implementation, but I just want to run this by Mathgroup to see if > >> this is to be expected, or maybe if I'm doing something suboptimal in > >> Mathematica. Here's the code that I'm running: > > >> grid1dc[x_, > >> y_] = (With[{d = 0.1, NN = 50}, > >> Sum[Re[N[ > >> d BesselJ[1, 2 Pi d Sqrt[m^2 + n^2]]/ > >> Sqrt[m^2 + n^2 + 10^-7]] Exp[ > >> I 2.0 Pi (m x + n y)]], {m, -NN, NN, 1}, {n, -NN, NN, 1= }]= > ]) // > >> N > > >> and > > >> gridres1da = > >> With[{delta = 0.5, xlim = 2.5, ylim = 2.5}, > >> Table[{x, y, grid1dc[x, y]}, {x, -xlim, xlim, delta}, {y, -ylim, > >> ylim, delta}]] > > >> Java implementation uses Colt and Apache common math libraries for the > >> Bessels and complex numbers, uses a double for loop, and consistently > >> runs 15-20 times faster. > > >> --Leo > >

**References**:**Mathematica 20x slower than Java at arithmetic/special functions, is***From:*Leo Alekseyev <dnquark@gmail.com>