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Asked: 2026-05-27T10:50:25+00:00

With Mathematica 8.0.1.0, I have used FindRoot[] to identify the intersection of two 2

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With Mathematica 8.0.1.0, I have used FindRoot[] to identify the intersection of two 2 pdf functions.

But if I have pdf functions that intersect at more than one point, and I have the upper limit of the x axis range beyond the second intersection, FindRoot[] only returns the second intersection.

pdf1 = 1/x 0.5795367855565214` (E^(
  11.170058830053032` (-1.525439351903338` - Log[x]))
   Erfc[1.6962452696714152` (-0.5548887795964352` - Log[x])] + 
 E^(1.2932713057519` (2.60836043407439` + Log[x]))
   Erfc[1.6962452696714152` (2.720730943938539` + Log[x])]);

pdf2 = 1/x 0.4648445097126269` (E^(
  5.17560914275408` (-2.5500941338198615` - Log[x]))
   Erfc[1.7747318880142482` (-2.139288893723375` - Log[x])] + 
 E^(1.1332542415053757` (3.050849516581922` + Log[x]))
   Erfc[1.7747318880142482` (3.1407996592474956` + Log[x])]);

Plot[{pdf1, pdf2}, {x, 0, 0.5}, PlotRange -> All]   (* Shows 1st intersection *)
Plot[{pdf1, pdf2}, {x, 0.4, 0.5}, PlotRange -> All] (* Shows 2nd intersection *)

{x /. FindRoot[pdf1 == pdf2, {x, 0.00001, 0.5}],
x /. FindRoot[pdf1 == pdf2, {x, 0.00001, 0.4}]}

enter image description here
enter image description here

The above plots show the issue. When plotted they intersect at two points:

{0.464719, 0.0452777}

respectively.

As I can’t know before hand if I’ll have a second intersection and I don’t know where it might fall on the x axis if I did, can anyone suggest a way to have FindRoot[] only return the first intersection rather than the second?

If not, can anyone suggest another way to go about it?

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1 Answer

  1. Editorial Team

    With FindRoot[], you can only get a single root for a given starting point. Iterating through different options is cumbersome and you might not even get the desired result for certain edge cases unless you hit upon the right choice of starting point.

    In this case, something like NSolve or Reduce might be a better option. If you know that your expressions decay, using a reasonable upper bound for possible values of x, you can use the following, which is pretty quick and will give you all roots.

    NSolve[{pdf1 == pdf2, 0 < x < 1}, x] // Timing
Out[1]= {0.073495, {{x -> 0.0452777}, {x -> 0.464719}}}
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