How to return the value of Automatic when it is used in a Mathematica function?

Question

If I have a function, for example:

p1 = Plot[Evaluate[S[t]*10^-12 /. sol], {t, 20 10^-9, 49 10^-9}]

and this function is setting options->Automatic so the function arguments are implicitly:

p1 = Plot[Evaluate[S[t]*10^-12 /. sol], {t, 20 10^-9, 49 10^-9}, 
          PlotPoints->Automatic, MaxRecursion->Automatic]

how do I get the value to which Automatic has been set?

Answer 1

Another option is to use Trace to find occurrences of the relevant rules which appear during evaluation:

Trace[
 Plot[Sin[t], {t, 0, 2 Pi}],
 HoldPattern[PlotPoints | MaxRecursion -> _],
 TraceInternal -> True] // Flatten // Union

{MaxRecursion -> 6, MaxRecursion -> Automatic, PlotPoints -> 50, PlotPoints -> Automatic}

Or you can set "Verbose" -> True in the system visualisation options to get lots of internal information as the plot is created:

SetSystemOptions["VisualizationOptions" -> {"Verbose" -> True}]
Plot[Sin[t], {t, 0, 2 Pi}]

(0.): >>>>>>>>>>>>>>> Plot <<<<<<<<<<<<<<<

(0.): Expr to Evaluate:

{Sin[t],{{t,0,2π}, Mesh->None, Exclusions->Automatic, PlotPoints->50, MaxRecursion->6, Filling->None, ColorFunction->...

Answer 2

$Version

"10.0 for Mac OS X x86 (64-bit) (September 10, 2014)"

From the documentation at http://reference.wolfram.com/language/tutorial/Options.html the default for PlotPoints is 50.

To determine the default for MaxRecursion, let

f[x_] = Product[x + n (-1)^n, {n, -4, 5}] E^(-x^2/2);

The number of points for the Plot of f[x] with the defaults for PlotPoints and MaxRecursion is

auto = Cases[Plot[f[x], {x, -5, 4}, PlotRange -> All], 
   Line[pts_] :> Length[pts], Infinity][[1]]

1154

With the default PlotPoints and varying MaxRecursion check for the same number of points:

Select[
 Cases[
     Plot[f[x], {x, -5, 4},
      PlotRange -> All,
      MaxRecursion -> #],
     Line[pts_] :> {#, Length[pts]},
     Infinity][[1]] & /@ Range[15],
 #[[2]] == auto &]

{{6, 1154}}

Hence, the default MaxRecursion is 6

Answer 3

When all else fail, i.e. Options[p1] or AbsoluteOptions[p1, PlotPoints], you can always grab the Line itself and find how many points it has:

 p1 = Plot[Sin[x], {x, 0, Pi}, PlotPoints -> Automatic]

 line = p1[[1, 1, 3, 2]];
 Graphics[line]

Length[line[[1]]]

So, 259 points are used in this case.

 ListPlot[line[[1]], Mesh -> All, PlotStyle -> Red]

You can see that M oversamples where the function changes most, as expected. ps. this is not the recommended way to find number of points used, since internal data structure can change, and index used above can become invalid in future versions.