# [R] Find Crossover Points of Two Spline Functions

Ben Bolker bbolker at gmail.com
Mon Sep 28 18:17:59 CEST 2015

```Dario Strbenac <dstr7320 <at> uni.sydney.edu.au> writes:

>
> Good day,
>
> I have two probability densities, each with a function determined
> by splinefun(densityResult[['x']],
> densityResult[['y']], "natural"), where densityResult is the
> output of the density function in stats.
> How can I determine all of the x values at which the densities cross ?
>

My initial thought was this is non-trivial, because the two densities could
cross (or nearly-but-not-quite cross) at an unlimited number of points.
I thought it would essentially boils down to "how do I find all
the roots of an arbitrary (continuous, smooth) function?

However, after thinking about it for a few more seconds I realize
that at least the functions are piecewise cubic.  I still don't see
a *convenient* way to do it ... if the knots were all coincident between
the two densities (maybe you could constrain them to be so?) then you
just have a difference of cubics within each segment, and you can use
polyroot() to find the roots (and throw out any that are complex or
don't fall within the segment).
If the knots are not coincident it's more of a pain but you should
still be able to do it by considering overlapping segments ...

```