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

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  • Subject: [mg115891] Re: Mathematica 20x slower than Java at arithmetic/special functions, is
  • From: Leo Alekseyev <dnquark at>
  • Date: Mon, 24 Jan 2011 05:56:56 -0500 (EST)
  • References: <>

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.


On Mon, Jan 24, 2011 at 4:14 AM, Vivek J. Joshi <vivekj at> wrote:
> Without going into too much detail, a simple compilation of the function 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}]];//AbsoluteTiming
> {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 special
>> functions.  In my particular example, I am simply evaluating a sum of
>> 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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