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Re: Fully vectorized system of ODE's - any advantage of C?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg121967] Re: Fully vectorized system of ODE's - any advantage of C?
*From*: Oliver Ruebenkoenig <ruebenko at wolfram.com>
*Date*: Fri, 7 Oct 2011 04:49:45 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
On Wed, 5 Oct 2011, DmitryG wrote:
> On Oct 4, 2:44 am, DmitryG <einsch... at gmail.com> wrote:
>> Dear All,
>>
>> I was working on solving systems of a big number of ODE's with
>> vectorization and compilation in C and I have received a great support
>> in this forum. The speed crucially depends on the details of
>> programming that not always can be detected by an average user (such
>> as myself).
>>
>> I have realized that my vectorized codes were not yet completely
>> vectorized because calls have been made to components of tensors. Then
>> I've got an idea of a code that is completely vectorized, for a model
>> of classical spins that is one of the subjects of my work. Here is the
>> problem that, I believe, can be of interest to many Mathematica
>> users.
>>
>> Dynamics of a chain of classical spins (described by three-component
>> vectors of fixed length) precessing in the effective field created by
>> neighbors
>>
>> ds[i]/dt==s[i]\[Cross]H[i]; H[i]=s[i-1]+s[i+=
> 1];
>> i=1,2,..., NSpins
>>
>> Both the vector product and interaction with neighbors in H[i] can be
>> very efficiently described by the RotateRight command. Putting all
>> three components of all NSpins spins into the double list
>>
>> s={ {s[x,1],s[x,2],..s[x,NSpins]}, {s[y,1],s[y,2],..s[y,NSpins]},
>> {s[z,1],sz,2],..s[z,NSpins]} },
>>
>> one can construct the code
>>
>> ************************************************************
>> (* Runge-Kutta-4 routine *)
>> ClearAll[makeCompRK4, makeCompRK5]
>> makeCompRK4[f_] :=
>> Compile[{{x0, _Real, 2}, {t0}, {tMax}, {n, _Integer}},
>> Module[{h, K1, K2, K3, K4, SolList, x = x0, t}, h = (tMax - t0)/n=
> ;
>> SolList = Table[x0, {n + 1}];
>> Do[t = t0 + k h;
>> K1 = h f[t, x];
>> K2 = h f[t + (1/2) h, x + (1/2) K1];
>> K3 = h f[t + (1/2) h, x + (1/2) K2];
>> K4 = h f[t + h, x + K3];
>> x = x + (1/6) K1 + (1/3) K2 + (1/3) K3 + (1/6) K4;
>> SolList[[k + 1]] = x, {k, 1, n}];
>> SolList](*,CompilationTarget->"C"*),
>> CompilationOptions -> {"InlineCompiledFunctions" -> True},
>> "RuntimeOptions" -> "Speed"]
>>
>> (* Defining equations *)
>>
>> RHS = Function[{t, s},
>> RotateRight[
>> s, {1, 0}] (RotateRight[s, {2, 1}] + RotateRight[s, {2, -1}])=
> -
>> RotateRight[
>> s, {2, 0}] (RotateRight[s, {1, 1}] + RotateRight[s, {1, -1}])=
> ];
>>
>> cRHS = Compile[{t, {s, _Real, 2}},
>> RotateRight[
>> s, {1, 0}] (RotateRight[s, {2, 1}] + RotateRight[s, {2, -1}])=
> -
>> RotateRight[
>> s, {2, 0}] (RotateRight[s, {1, 1}] + RotateRight[s, {1, -1}])=
> ];
>>
>> (*Compilation*)
>> tt0 = AbsoluteTime[];
>> Timing[RKComp = makeCompRK4[RHS];]
>> AbsoluteTime[] - tt0
>>
>> (* Initial condition *)
>> x0 = { Join[{1}, Table[0, {i, 2, NSpins}], {0}],
>> Join[{0}, Table[0, {i, 2, NSpins}], {0}],
>> Join[{0},
>> Table[1, {i, 2,
>> NSpins}], {0}] }; (* Padding with a zero spin at the NSpins=
> +1 \
>> position *)
>>
>> (* Parameters *)
>> NSpins = 120; t0 = 0; tMax = 150; n = 1000;
>>
>> (* Solving *)
>> tt0 = AbsoluteTime[];
>> Sol = RKComp[x0, t0, tMax, n];
>> AbsoluteTime[] - tt0
>>
>> Print["Compilation: ", Developer`PackedArrayQ@Sol]
>>
>> (* Plotting *)
>> tList = Table[1. t0 + (tMax - t0) k/n, {k, 0, n}];
>> si\[Alpha]List[\[Alpha]_, i_] :=
>> Table[Sol[[k]][[\[Alpha], i]], {k, 0, n}];
>> s1List = Transpose[{tList, si\[Alpha]List[3, 1]}];
>> s2List = Transpose[{tList, si\[Alpha]List[3, 40]}];
>> s3List = Transpose[{tList, si\[Alpha]List[3, 80]}];
>> s4List = Transpose[{tList, si\[Alpha]List[3, NSpins]}];
>> ListPlot[{s1List, s2List, s3List, s4List},
>> PlotStyle -> {Blue, Green, Red, Orange}, PlotRange -> All]
>>
>> *************************************************************************=
> ** **********
>>
>> The code is very short and runs faster than all my previous codes for
>> the same problem. There are some questions, however:
>>
>> 1) There is no difference in speed between uncompiled RHS and compiled
>> cRHS.
>>
>> 2) On my laptop (Windows 7, 64 bit), compilation in Mathematica
>> results in 0.070 execution time, whereas compilation in C (Microsoft
>> Visual C++) is longer, 0.075. In all previous not fully vectorized
>> versions of the code compiling in C gave a speed advantage by a factor
>> 2-3. Is it a perfect code that cannot be improved by C or, for some
>> reason, compilation in C does not work and the system returns to the
>> Mathematica compiler?
>>
>> Oliver mentioned earlier that CopyTensor in the loop in the compiled
>> code leads to slowdown. Here I do have CopyTensor in the loop. Could
>> it be moved outside??
>>
>> I will be very greatful for all comments, as usual.
>>
>> Dmitry
>
> I have run this program many times with
>
> RKComp = makeCompRK4[RHS];
>
> and
>
> RKComp = makeCompRK4[cRHS];
>
> and also with compilation in C or in Mathematica, and I see that there
> is no execution time difference that would stand out of fluctuations.
> Thus I suspect that in the code above compilation in C does not work
> and there is a fallback to the Mathematica compiler.
>
> Dmitry
>
>
I suspect that there is not overhead until the code reaches the library
function that is eventually evaluated in either C or the WVM. A profile
would show this.
Oliver
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