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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
>
>



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