allcrossings

Locate all intersections of a pair of functions f1 and f2, on a finite domain

ALLCROSSINGS is a rootfinding tool , that is used to locate all points where a pair of functions cross in the plane. It solves for the locations x, where f1(x)==f2(x). As a simple example, here is the use of allcrossings, on the problem sin(x)==cos(x). We expect those solutions happen at the points (n+1/4)*pi. The search was done on the interval [-10,10].

xloc = allcrossings(@sin,@cos,[-10,10])
xloc =
  -8.6394  -5.4978  -2.3562   0.7854    3.927   7.0686

If I now divide by pi, we should expect to see solutions that are all off from an integer by exactly 1/4.

xloc/pi
ans =
    -2.75    -1.75    -0.75    0.25     1.25     2.25

The tool initially samples the supplied interval at equally spaced points, locating sub-intervals where a crossing must occur, then it uses fzero on that bracketed interval.

ALLCROSSINGS can also be used to identify crossings of a function with a constant. In this example, I used it to locate the points where y=2*x+tan(x) crosses the line y==1, on the interval [0,10].

xloc = allcrossings(@(x) 2*x + tan(x),1,[0,10])
xloc =
  0.32919   1.5708   1.9113   4.7124   4.8274    7.854   7.9213

As one should expect though, the function here has singularities, where while it technically crosses the constant line y==1, they are not actually intersections. Those points are easily enough identified though, since wheere the function passes through a discontinuity, we see those points are clearly identified.

fun = @(x) 2*x + tan(x);
fun(xloc)
ans =
   1  -1.2093e+15  1  3.5109e+14   1   2.5914e+14   1

I intentionally do not try to remove those locations, since they could technically be called a crossing point.