       Re: Vector Runge-Kutta ODE solver with compilation?

• To: mathgroup at smc.vnet.net
• Subject: [mg116692] Re: Vector Runge-Kutta ODE solver with compilation?
• From: DmitryG <einschlag at gmail.com>
• Date: Thu, 24 Feb 2011 06:22:09 -0500 (EST)
• References: <201102231024.FAA09857@smc.vnet.net> <ik2qs3\$avl\$1@smc.vnet.net>

```On 23 Feb., 06:25, Oliver Ruebenkoenig <ruebe... at wolfram.com> wrote:
> On Wed, 23 Feb 2011, DmitryG wrote:
>
> [....]
>
>
>
>
>
>
>
>
>
> > Dear Oliver and All interested,
>
> > Today I tried to apply the above very efficient vector RK4 method with
> > compilation to my real problems that I was solving with NDSolve
> > before. Unfortunately, it has proven to be more complicated than it
> > seemed. To illustrate the difficulties, I will use the same working
> > model as above and will try to generalize it. So, the working code is
>
> > *************************************************
>
> > (* Definition of the RK4 solver *)
> > makeCompRK[fIn_] :=
> >  With[{f = fIn},
> >   Compile[{{x0, _Real, 1}, {t0, _Real}, {tMax, _Real}, {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"]];
>
> > (* Calculation *)
> > F = Function[{t, x}, {x[], 1 - x[] + 4/x[]^3}];
> > RK4Comp = makeCompRK[F];
> > x0 = {1.0, 1.0};   t0 = 0;    tMax = 200;   n = 5000;
> > tt0 = AbsoluteTime[];
> > Sol = RK4Comp[x0, t0, tMax, n];
> > AbsoluteTime[] - tt0
> > Developer`PackedArrayQ@Sol
>
> > (* Plotting *)
> > nTake = 200;
> > ListLinePlot[Take[Sol, nTake], PlotMarkers -> Automatic, PlotStyle ->
> > {Red}]  (* Trajectory *)
>
> >  (* Time dependences *)
> > x1List = Table[{1. t0 + (tMax - t0) k/n, Sol[[k + 1, 1]]}, {k, 0, n}]=
;
> > x2List = Table[{1. t0 + (tMax - t0) k/n, Sol[[k + 1, 2]]}, {k, 0, n}]=
;
> > ListLinePlot[{Take[x1List, nTake], Take[x2List, nTake]},
> > PlotMarkers -> Automatic, PlotStyle -> {Blue, Green}]
> > *************************************************************************** > > **************************
>
> > In this example, the vector of ODE's RHS' is defined by
>
> > F = Function[{t, x}, {x[], 1 - x[] + 4/x[]^3}];
>
> > In real life, this is something more complicated, a result of a many-
> > step calculation. One can try to generalize the code as
>
> > RHS={x[], 1 - x[] + 4/x[]^3};
> > F = Function[{t, x}, RHS];
>
> Dmitry,
>
> your friend is - in princial -
>
> F = Function[{t, x}, Evaluate[RHS]]
>
> That has a small problem that we used Part in the RHS and Part gives a
> message that you try to access elements that the _symbol_ x does not have.
>
> One way to work around this is
>
> RHS = Quiet[{x[], 1 - x[] + 4/x[]^3}];
> F = Function[{t, x}, Evaluate[RHS]];
>
> If you evaluate F you'll see that the RHS in now in the function body. If
> you look at the F you posted, there the body is RHS, which is not what
> you want. All this hast to do with
>
> Attributes[Function]
>
> HoldAll.
>
> Oliver
>
>
>
>
>
>
>
>
>
> > (that should be equivalent to the above) and then substitute RHS by
> > whatever you have for your problem. However, Mathematica complains and
> > produces crap results. My suspicion is that it is only F =
> > Function[{t, x}, RHS]; that is injected into the compiled RK4 code
> > whereas RHS={x[], 1 - x[] + 4/x[]^3}; is not injected.
> > However, I cannot be sure because it was never explained.
>
> > Further idea is to define F as a procedure and put everything inside
> > it. A toy example is
>
> > F = Function[{t, x}, RHS = {x[], 1 - x[] + 4/x[]^3}; RHS];
>
> > This does not compile, although then retreating to the non-compilation
> > mode produces correct results. My experiments show that you can only
> > use a single expression in the body of F that does not use any
> > definitions. For instance,
>
> > F = Function[{t, x}, Table[-x[[i]]^i, {i, 1, 2}]];
>
> > compiles and works fine but this is, again, no more than a toy
> > example.
>
> > To summarize, I am very excited by the prospect to speed up my
> > computations by a factor about 100 but I don't see how to generalize
> > the approach for real-life problems.

Thank you, Oliver, this was a great help.

Now, I am trying to introduce sampling of the results to reduce the
number of output points. This is also very important. However, there
is a new problem. Here is the code:

******************************************************************************
(* Vector RK4 routine *)
makeCompRK[fIn_]:=With[{f=fIn},Compile[{{x0,_Real,1},{t0,_Real},
{ttMax,_Real},{n,_Integer},{nsamp,_Integer}},
Module[{h,K1,K2,K3,K4,SolList,x=x0,t,ksamp},
h=(ttMax-t0)/n;
SolList=Table[x0,{nsamp+1}];
Do[t=t0+k h;        ksamp=k nsamp/n;
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;

If[FractionalPart[ksamp]==0,SolList[[ksamp+1]]=x]

,{k,1,n}];
SolList],CompilationTarget->"C"]];

(* Calculation *)
RHS=Quiet[{x[],1-x[]+4/x[]^3}]; (* Quiet to suppress
complaints *)
F=Function[{t,x},Evaluate[RHS]];  (* Evaluate to use actual form of
RHS *)
RK4Comp=makeCompRK[F];
x0={1.0,1.0};   t0=0;    tMax;   NSteps=5000;  NSampling0;
tt0=AbsoluteTime[];
Sol=RK4Comp[x0,t0,tMax,NSteps,NSampling];
AbsoluteTime[]-tt0
Developer`PackedArrayQ@Sol
******************************************************************************

The problem is in the line

If[FractionalPart[ksamp]==0,SolList[[ksamp+1]]=x]

In the first execution, Mathematica complains

Compile::cpintlt: "ksamp\$+1 at position 2 of SolList\$[[ksamp\$+1]]
should be either a nonzero integer or a vector of nonzero integers;
evaluation will use the uncompiled function."

I do not understand this. Still Mathematica compiles and the results
are correct.

In the second execution, Mathematica complains more and does not
compile, although the results are still correct.

If I use the command

If[IntegerQ[kSampling],SolList[[kSampling+1]]=x

Mathematica complains again:

CompiledFunction::cfse: "Compiled expression False should be a machine-
size real number)."

does not compile, but the results are correct.

What is wrong here? I believe I am doing everything in a natural way.
But compiling is too tricky..

Dmitry

```

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