Telling Plot to style vector-valued black-box functions in Mathematica

Question

Suppose I write a black-box functions, which evaluates an expensive complex valued function numerically, and then returns real and imaginary part.

fun[x_?Ine         


        

Answer 1

If possible for your actual application, one way is to allow fun to take symbolic input in addition to just numeric, and then Evaluate it inside of Plot:

fun2[x_] := Module[{f = Sin[x]}, {Re[f], Im[f]}]

Plot[Evaluate[fun2[x + I]], {x, -1, 1}]

This has the same effect as if you had instead evaluated:

Plot[{-Im[Sinh[1 - I x]], Re[Sinh[1 - I x]]}, {x, -1, 1}]

Answer 2

One possibility is:

Plot[{#[[1]], #[[2]]}, {x, -1, 1}, PlotStyle -> {{Red}, {Blue}}] &@ fun[x + I]  

Edit

If your functions are not really smooth (ie. almost linear!), there is not much you can do to prevent the double evaluation process, as it will happen (sort of) anyway due to the nature of the Plot[] mesh exploration algorithm.

For example:

fun[x_?InexactNumberQ] := Module[{f = Sin[3  x]}, {Re[f], Im[f]}];
Plot[{#[[1]], #[[2]]}, {x, -1, 1}, Mesh -> All, 
   PlotStyle -> {{Red}, {Blue}}] &@fun[x + I]  

Answer 3

I don't think there's a good solution to this if your function is expensive to compute. Plot will only acknowledge that there are several curves to be styled if you either give it an explicit list of functions as argument, or you give it a function that it can evaluate to a list of values.

The reason you might not want to do what @belisarius suggested is that it would compute the function twice (twice as slow).

However, you could use memoization to avoid this (i.e. the f[x_] := f[x] = ... construct), and go with his solution. But this can fill up your memory quickly if you work with real valued functions. To prevent this you may want to try what I wrote about caching only a limited number of values, to avoid filling up the memory: http://szhorvat.net/pelican/memoization-in-mathematica.html