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Re: Fully vectorized system of ODE's - any advantage of C?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg121899] Re: Fully vectorized system of ODE's - any advantage of C?
*From*: DmitryG <einschlag at gmail.com>
*Date*: Thu, 6 Oct 2011 04:21:51 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <j6ea1b$lf2$1@smc.vnet.net> <j6h3gs$746$1@smc.vnet.net>
On Oct 5, 4:12 am, DmitryG <einsch... at gmail.com> 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 NSpi=
ns=
> +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
Another thing: I cannot use equations defined above in the standard
Mathematica way without compilation, using NDSolve. The problem is
that for x unspecified RotateRight[x,{...}] outputs x that is wrong
and yields to RHS=0.
Dmitry
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