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Re: Plotting vector-valued functions

Malcolm Boshier wrote:

>     I have a problem which is related to the recent thread about
> plotting lists of functions.  In the case when a vector-valued function
> is expensive or impossible to Evaluate before plotting, Plot apparently
> forces you to evaluate the function repeatedly at each value of the
> independent parameter.  This can be very inefficient.
>     As an example, suppose that f[z] returns the eigenvalues of a 5 x 5
> matrix which is a function of z.  

Ok, say, 

In[1]:= m[z_] = {{2, 1, 9, 5, 1}, {5, 5, 1, 3, 7}, {9, 4, 8, 7, 5}, 
    {3, 8, 1 + z, 8, 7}, {7 - 2*z, 2, 3, 8, 8}}; 

In[2]:= f[z_] := Eigenvalues[m[z]]

> In general this function cannot be evaluated without a value for z, 

Actually, it _can_ be evaluated as a function of z in terms of
(algebraic) Root objects.  Try entering f[z].

> so Plot[ Evaluate[f[z]], {z, zmin, zmax}] doesn't work.

This does work for me:

In[3]:= Plot[Evaluate[Re[f[z]]], {z, -10, 10}];

>     The only way around this that I have found is something like:
> Plot[{f[z][[1]],  f[z][[2]], f[z][[3]], f[z][[4]], f[z][[5]]}, {z, zmin,
> zmax}]
> which of course requires 5 evaluations of f[z] for each value of z.

Dynamic programming can be used to avoid this.  E.g., you could define

In[3]:= f[z_?NumericQ] := f[z] = Eigenvalues[m[z]]

Then, when f[z][[1]] is evaluated for a particular z, f[z] is recorded
and saved.

Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907  AUSTRALIA                   

            God IS a weakly left-handed dice player

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