Consider the following contour of intersections between two analytic expressions:
ContourPlot[ Sinh[x^2] (x^2 + Cosh[y^2] y^2) == y^2, {x, -1, 1}, {y, -2, 2}]
The above nicely visualizes the intersection set, and also implies that the implementation of ContourPlot
contains a very efficient algorithm that evaluates the intersection region numerically before plotting it. If I try to find numeric solutions to this problem, I fix i.e. x=1/2
and use something like FindRoot
to get y->0.290862
, or x=1/3
and y->0.117988
, always one point at a time. This is a very inconvenient way to approach this problem, since it also tends to jump whenever several intersections exist and numerically break down whenever the slope becomes large.
Surely, there must be a better way to evaluate the intersection region numerically? Is there a function in Mathematica that would return a convenient representation of the numeric intersection region? Or maybe one can write such a function efficiently? Thanks for any suggestion!
FindRoot
. See, for instance, here. This is actually a pretty common method; there should be many examples of this on this very site. $\endgroup$ – march Oct 3 '16 at 18:13