NDSolve/InterpolatingFunction and vectors

• To: mathgroup at smc.vnet.net
• Subject: [mg53657] NDSolve/InterpolatingFunction and vectors
• From: D Herring <dherring at at.uiuc.dot.edu>
• Date: Fri, 21 Jan 2005 06:37:09 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi all,

For numerous reasons (such as dot products), I would like to use NDSolve
with vector-valued functions.

For example, the Sine and Cosine could be defined as
soln=NDSolve[{
xx'[t]\[Equal]{{0,1},{-1,0}}.xx[t],xx[0]\[Equal]{{0},{1}}
},xx,{t,0,10}]

Then I have the multi-valued function
f[t_]=(xx/.soln[[1]])[t]

such that f[Pi] is roughly {{0},{-1}} as expected.

The trouble comes when trying to extract scalar values from f[t].
f[3.14][[1,1]] ~= 0 but f[t][[1,1]] throws an error.
Likewise, Plot[f[t],{t,0,10}]  pukes because f[t] doesn't return a
scalar when evaluated.

My current solution uses dot products to extract values.
{1,0}.f[t] doesn't error since it holds until f is evaluated.
Plot[{{1,0},{0,1}}.f[t],{t,0,10}] still bombs, but
Plot[{{1,0}.f[t],{0,1}.f[t]},{t,0,10}] works fine.

Can anyone suggest a better overall method for embedding dot products in
NDSolve?  The system I have looks something like