Plot the minimum of a list of functions

Question

This is my first question on any stack-exchange site and I'm also very new to using Mathematica software so please excuse/correct me if I mess up.

I would like to take a list of functions of a single variable {$f_1, f_2, f_3, \ldots, f_n$}, and plot the minimum value that each of these functions takes on at any point over an interval $[x_0, x_f]$.

So far this much is not so difficult (or has not been the difficult part for me to figure out). However, I would like for each of the intervals over which any function $f_i$ is the minimum to correspond to it's own color (like how Plot usually attributes different colors to different curves when you plot multiple functions) and further I'd like to include a legend that clearly denotes which function from the list corresponds to which color.

I can do the first part fairly easily using the following:

Z[x_] = Min[x, x^2, x^3, x^4, x^5]
Plot[Z[x], {x, 0, 2}, PlotLegends -> "Expressions"]

I would like something that would color the segment of the curve where $x^5$ is the minimum function in the list one color and the segment where $x$ is the minimum function in another color and denote the color representations on the legend on the right.

I can plot each of the functions separately with the color-coding system like so:

Plot[{x, x^2, x^3, x^4, x^5}, {x, 0, 2}, PlotLegends -> "Expressions"]

And in both cases the " PlotLegends -> "Expressions" " flag/option (not sure what the terminology is for Mathematica functions) does nicely generate a legend as I'd like it.

But I don't really have a sense of how to combine the two.

Thanks in advance for the help, and while I didn't find any other questions that might help answer (or partially answer) this question please correct me if I missed any.

Answer 1

New method

Inspired by your self-answer we can automate things as follows:

prep[fn_][a__] := If[# == fn[a], #] & /@ {a}

Now:

Plot[prep[Min][x, x^2, x^3, x^4, x^5], {x, 0, 2},
 BaseStyle -> {14, Thick},
 Frame -> True,
 PlotStyle -> {Red, Orange, Yellow, Green, Blue},
 Evaluated -> True
]

Although untested I presume this will work with PlotLegends. Note:

• prep is somewhat generalized so that it can work with other functions
• Evaluated -> True is used rather than Evaluate to keep x localized to the Plot
• This method doesn't work in version 9.0.0 due to a bug. (47981)

---

Old methods

It may surprise you to learn that Mathematica internally splits the Line on discontinuities (when using the default value Automatic for the Exclusions option), which allows you do use the post-processing method shown here (bottom), e.g.:

f[x_] := Min[x, x^2, x^3, x^4, x^5]

Module[{i = 1},
 Plot[f[x], {x, 0, 2}, PlotStyle -> Thick]
  /. x_Line :> {ColorData[1][i++], x}
]

Or Simon Woods's splitstyle:

splitstyle[styles__] := 
  Module[{st = Directive /@ {styles}}, {{Last[st = RotateLeft @ st], #}} &];

Plot[f[x], {x, 0, 2}, PlotStyle -> splitstyle[Red, Green], BaseStyle -> Thick]

Notes

I don't have the PlotLegends option in version 7, which I use, therefore I cannot test that aspect of the question. Pardon me for not mentioning that directly.

You asked for an explanation of these methods. They are similar, yet work different. Both rely on the operation of the Exclusions mechanism of Plot. When a discontinuity is found a new Line primitive is created within the Graphics expression that is produced by Plot. The first method works by replacing (see ReplaceAll) each Line expression with a {style, Line} pair. (style is drawn from an arbitrarily selected ColorData scheme.)

The second method relies on a clever construction and the behavior of the PlotStyle option when it is given a function as its value. The function generated by splitstyle uses this method to cycle between given styles (not used in this example, but useful elsewhere). It also produces a {style, Line[ . . . ]} pair, and this expression is inserted by Plot itself rather than with post-processing.

Answer 2

Simple solution using RegionFunction:

f[x_] := {x, x^2, x^3, x^4, x^5};
Plot[Evaluate@f[x], {x, 0, 2}, PlotLegends -> "Expressions", 
 RegionFunction -> Function[{x, y}, y == Min[f[x]]], PlotPoints -> 20, 
 PlotRange -> All, PlotStyle -> ColorData[35, "ColorList"]]

One limitation is that you get overshooting unless you fiddle with the number of PlotPoints.

The only reason I changed the PlotStyle is because the first and fifth colours are hard to tell apart in the default colour scheme.

Answer 3

flist = Table[BesselJ[n, x], {n, 4}];
pieces = Table[ConditionalExpression[f, f == Min[flist]], {f, flist}]; (* thanks Rahul *)
pltstyls = Join[#, Directive[{#, Thickness[.01], Dashed}] & /@ #] &[
            ColorData[1, "ColorList"][[;; Length@flist]]];
lgndlbls = Join[#, StringJoin["piece ", #] &/@ (ToString /@ #)]&[TraditionalForm /@ flist];


Plot[Evaluate@Join[flist, pieces], {x, 0, 10},
     Filling -> Thread[Range[Length@flist] -> Axis], ImageSize -> 500,
     PlotStyle -> pltstyls, PlotLegends -> lgndlbls]

Update: Generalizing to arbitrary list of functions flist and functionals on flist:

 foo = Module[{pieces =Table[ConditionalExpression[f, f == #[#2]], {f, #2}], 
        styles = Join[#, Directive[{#, Thickness[.01], Dashed}] & /@ #] &[
                    #3[[;; Length@#2]]],
       lgndlbls =  Join[#, StringJoin["piece ", #] & /@ (ToString /@ #)] &[
                    TraditionalForm /@ #2]},
 Plot[Evaluate@Join[#2, pieces], {x, 0, 10}, 
        Filling -> Thread[Range[Length@#2] -> Axis], ImageSize -> 500, 
        PlotStyle -> styles, PlotLegends -> lgndlbls]] &;

 foo[RankedMin[#, 2] &, flist, ColorData[1, "ColorList"]]

Answer 4

I have somewhat devised a solution that does what I'm looking for but requires manually generating a list of functions that I'd like to avoid.

Z[x_] = Min[x, x^2, x^3, x^4, x^5]
g1[x_] = Piecewise[{{x, x == Z[x]}}]
g2[x_] = Piecewise[{{x^2, x^2 == Z[x]}}]
g3[x_] = Piecewise[{{x^3, x^3 == Z[x]}}]
g4[x_] = Piecewise[{{x^4, x^4 == Z[x]}}]
g5[x_] = Piecewise[{{x^5, x^5 == Z[x]}}]
Plot[{g1[x], g2[x], g3[x], g4[x], g5[x]}, {x, 0, 2}, PlotRange -> All, PlotLegends -> "Expressions"]

Produces the following output

Some way to automate this would be much more preferred. In general I might want to err on the side of using Mr.Wizard's solution for my use case as I intend to do this repeated for lists of many functions at a time.

Further it might be nice for the expressions in the legend to include the original definitions of the functions $f$, but this is something that can be adjust manually fairly easily.