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Re: How to avoid repeated calculation in NDSolve ?

• To: mathgroup at smc.vnet.net
• Subject: [mg117008] Re: How to avoid repeated calculation in NDSolve ?
• From: Oliver Ruebenkoenig <ruebenko at wolfram.com>
• Date: Mon, 7 Mar 2011 05:49:12 -0500 (EST)
• References: <201103061042.FAA15467@smc.vnet.net>

On Sun, 6 Mar 2011, t. shim wrote:

> Hi,
>
> As the samplke list below,  the rhs of equations in NDSolve may have
> many repeated appearance of  the same expression such as (\[Mu][t]^2 +
> \[Nu][t]^2) in the denominator.
> In Fortran, we first write r = (\[Mu][t]^2 + \[Nu][t]^2) and in what
> follows use r insted of (\[Mu][t]^2 + \[Nu][t]^2).
> Tell me if such can be applied also in mathematica.
>
> Regards, t. shim
>
> NDSolve[
> {p\[Mu]'[t] =
> 1/2 (2 \[Mu][
>      t] (1/2 (
>        p\[Mu][t]^2 + p\[Nu][t]^2-4)/(\[Mu][t]^2 + \[Nu][t]^2) -
> (1 - \[Gamma])/2 (\[Mu][t] p\[Mu][t] - \[Nu][t] p\[Nu][t])^2/ (\[Mu]
> [t]^2 + \[Nu][t]^2)^2) +
> (1 - \[Gamma]) 1/(\[Mu][t]^2 + \[Nu][t]^2)^2 \[Nu][t] (\[Mu][t] p\[Mu]
> [t] -
> \[Nu][t] p\[Nu][t]) (\[Nu][t] p\[Mu][t] + \[Mu][t] p\[Nu][t])),
> p\[Mu]'[t] = (mu, nu interchanged)
> ...
>
>

Hi,

if you look at, say,

test = {p\[Mu]'[t] =
    1/2 (2 \[Mu][
         t] (1/2 (p\[Mu][t]^2 + p\[Nu][t]^2 -
              4)/(\[Mu][t]^2 + \[Nu][t]^2) - (1 - \[Gamma])/
            2 (\[Mu][t] p\[Mu][t] - \[Nu][t] p\[Nu][t])^2/(\[Mu][
                 t]^2 + \[Nu][t]^2)^2) + (1 - \[Gamma]) 1/(\[Mu][
              t]^2 + \[Nu][t]^2)^2 \[Nu][
         t] (\[Mu][t] p\[Mu][t] - \[Nu][t] p\[Nu][t]) (\[Nu][t] p\[Mu][
            t] + \[Mu][t] p\[Nu][t]))}


and then at

Experimental`OptimizeExpression[test]

you see the optimized expr. that NDSolve uses. So, NDSolve does this 
automatically.

Oliver

• References:
  • How to avoid repeated calculation in NDSolve ?
    • From: "t. shim" <tokuzoshimada3@gmail.com>