Numerical contour plot of compiled function

Question

I have come across a problem in Mathematica when I have tried to plot the contours of a compiled function.

f  = Compile[{{x}, {y}}, x^2 + y^2];
ContourPlot[f[x, y] == 0.1, {x, y} \[Element] Disk[{0, 0}, 1]]

this works perfectly but suppose I want to only plot a single contour:

ContourPlot[f[x, y] == 0.1, {x, y} \[Element] Disk[{0, 0}, 1]]

this then returns the error

but still plots the contour correctly. Evaluating f[x, y] == 0.1 gives x^2 + y^2 == 0.1 along with the same error message as the output so it's clear that the problem is that in order to plot the contour specified Mathematica requires the equation for the contour to be in symbolic form suggesting that ContourPlot is using a symbolic method to solve the problem.

However this is no good for anything difficult. I have a function longFunc[x,y] which is an extremely complicated expression and hence must be compiled and evaluated numerically. When I do a simple ContourPlot it works perfectly but if I specify a contour longFunc[x,y]==0.5 it falls back to a symbolic method and doesn't return an answer for a very long time.

So the question is: Is there an option within ContourPlot to force Mathematica to search for a solution to the equality using a numerical method?

Thank you!

Related:

ContourPlot is slow and unwieldy and generates a large-data graphic

Plot compiled function with LogLinearPlot

Answer 1

Ok I figured it out. You specify the contours you want through the Contours option of the function, surprisingly.

f = Compile[{{x}, {y}}, x^2 + y^2];
ContourPlot[f[x, y], {x, y} \[Element] Disk[{0, 0}, 1], 
 Contours -> {0.1}, ContourShading -> False]

Works beautifully. If you specify an equality in the argument ContourPlot defaults to symbolic solving which is usually way too slow.

Answer 2

Try

f = Compile[{{x, _Real}, {y, _Real}}, x^2 + y^2];
ContourPlot[f[x, y] == 0.1, {x, y} \[Element]Disk[{0, 0}, 1]]

Answer 3

The message you are getting means that ContourPlot internally is calling f with symbolic arguments contrary to its documentation.

ContourPlot has attribute HoldAll, and evaluates the f_i and g_i only after assigning specific numerical values to x and y.

Nevertheless, many functions expect to be able to evaluate user-supplied functions symbolically. There is an idiomatic way to prevent these calls: tell the pattern matcher that the arguments must be numeric.

Clear[f, fCompiled];
fCompiled = Compile[{{x}, {y}}, x^2 + y^2];
f[x_?NumericQ, y_?NumericQ] := fCompiled[x, y];
ContourPlot[f[x, y] == 0.1, {x, y} \[Element] Disk[{0, 0}, 1]]

This produces no diagnostics.

There are two confusable conditions, ?NumberQ and ?NumericQ. Since NumberQ[Pi] == False, NumericQ[Pi] == True, and fCompiled[Pi,Pi] == 19.739208802178716`, NumberQ is too strict. Use NumericQ.

Answer 4

Here are a couple more ideas for displaying the contour at $z=1/10$

You could also define a helper function g that forces Plot3D to work with numeric values. Note: this will give a symbolic rather than a numeric tooltip for the contour.

g[x_?NumericQ, y_?NumericQ] := f[x, y];
ContourPlot[g[x, y] == 1/10, {x, -1, 1}, {y, -1, 1}, ContourShading -> False]

You could also show the contour on the 3D surface defined by f:

Plot3D[f[x, y], {x, y} ∈ Disk[],
  MeshFunctions -> {#3 &},
  Mesh -> {{{1/10, Directive[Black, AbsoluteThickness[3.5]]}}},
  BoxRatios -> {1, 1, 1/3},
  ColorFunction -> (White &),
  Lighting -> "Neutral"]