       Re: How to use FindMaximum with a parameter passed to NDSolve??

• To: mathgroup at smc.vnet.net
• Subject: [mg129140] Re: How to use FindMaximum with a parameter passed to NDSolve??
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Tue, 18 Dec 2012 02:34:46 -0500 (EST)
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```Since pendulum uses NDSolve it can only be evaluated for a numeric
argument; consequently, its definition should be restricted to numeric
arguments. You will also need to use a higher precision than machine
precision.

eqns = Rationalize[{
fi''[t] + 9.8 Cos[long] Sin[fi[t]] +
.5 fi'[t] == 0, fi == Pi/2, fi' == 0}, 0];

pendulum[long_?NumericQ] := First[fi /. NDSolve[
eqns, fi, {t, 0, 50}, WorkingPrecision -> 25]]

total[dynamics_, p_] := dynamics[31/10] + p

max = FindMaximum[{total[pendulum[long], long]},
{long, 0.01`30}, WorkingPrecision -> 30];

max /. x_?NumericQ :> Round[x, 10.^-6]

{13.3595, {long -> 9.81034}}

Bob Hanlon

On Mon, Dec 17, 2012 at 2:58 AM, JBB <barandiaran.juan at gmail.com> wrote:
> Hello,
>
> This is probably a simple sintax question, but could somebody tell me how can I use the FindMaximum function when the variable used has to be used in an internal NDSolve??
>
> A simplified version of some code to show the error is as follows:
>
> pendulum[long_] :=
>  First[fi /.
>    NDSolve[{fi''[t] + 9.8 Cos[long] Sin[fi[t]] + .5 fi'[t] == 0,
>      fi == Pi/2, fi' == 0}, fi, {t, 0, 50}]]
>
> total[dynamics_, p_] := dynamics[3.1] + p
>
> FindMaximum[{total[pendulum[long], long ] }, {long, 0.01}]
>
> I get the following error:
>
> In:= FindMaximum[
>  Evaluate[total[pendulum[long], long ] ], {long, 0.01}]
>
> During evaluation of In:= NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.`. >>
>
> During evaluation of In:= ReplaceAll::reps: {NDSolve[{9.8 Cos[long] Sin[fi[<<1>>]]+0.5 (fi^\[Prime])[t]+(fi^\[Prime]\[Prime])[t]==0,fi==\[Pi]/2,(fi^\[Prime])==0},fi,{t,0,50}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>
>
> During evaluation of In:= FindMaximum::nrnum: The function value -0.01-fi[3.1] is not a real number at {long} = {0.01}. >>
>
> Out= FindMaximum[long + fi[3.1], {long, 0.01}]
>
> If I understand correctly, probably because the variable "long" has not been assigned a value by FindMaximum  before calling the internal NDSolve.
>
> Is there a simple way of doing this?
> In my real case the functions are longer but the essence of the problem is the same: I have to find the maximum of some function in which the variables are parameters of other functions including a NDSolve.
>
> Thanks for any hint,
>
> JBB

```

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