# 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.  In general this function cannot be
> evaluated without a value for z, so
> Plot[ Evaluate[f[z]], {z, zmin, zmax}] doesn't work.
>     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.
>     It seems that unless the head of the first argument to Plot is List,
> Plot assumes that it will evaluate to a real number and returns with an
> error if it later finds that it doesn't.  Why can't Plot trust the user
> long enough to discover that the function will evaluate to a list?
> Thanks for any solutions or explanations, Malcolm

Malcolm

I'll take

f[z_] := Chop[Eigenvalues[Table[Sin[(n  m)z],{n,5},{m,5}]]]

(symmetric matrix, so real eignvalues, but could be spurious small
imaginary parts, so use Chop)

Plot[{f[z][[1]],f[z][[2]],f[z][[3]],f[z][[4]],f[z][[5]]},
{z,0,1}]//Timing//First

11.3833 Second

One problem is that Plot[{f,g}, {x,xmin, xmax}] seems to compute the
lines for f and g separately and then show them together (this is so
even if we set PlotDivision -> 1).This means that the adaptive sampling
usually chooses different point for the two functions. So remembering
everything that was found for f might not help in calculating for g.

For example,

Plot[{0,Sin[10x]},{x,0,1}, PlotDivision->1]

The number of points used for the two lines is

Cases[%,ln_Line:> Length@@ln, Infinity]

{25,34}

Here is a faster way, but it is not adaptive and so gives a poorer plot.

Show[Graphics[Line/@
Axes->True]//Timing//First

2.46667 Second

But we see that the first plot seems to have lost some features near z =
.36. Let's see -

Plot[{f[z][[1]],f[z][[2]],f[z][[3]],f[z][[4]],f[z][[5]]}, {z,0,1},
PlotPoints -> 200]//Timing//First

21.6833 Second

--
Allan Hayes
Training and Consulting
Leicester, UK
hay@haystack.demon.co.uk
http://www.haystack.demon.co.uk
voice: +44 (0)116 271 4198
fax: +44 (0)116 271 8642

```

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