False Position Method

It is possible to use a modification of false position for finding minimums of functions to find the root (if it is indeed one).

The idea is that three points $x_mathrm L<x_mathrm M<x_mathrm R$ are given so that $f(x_mathrm M)<f(x_mathrm{L~and~R})$.

We then generate a new point $x_mathrm N$ by applying false position to the points $(x_mathrm L,pm f(x_mathrm L))$ and $(x_mathrm R,mp f(x_mathrm R))$, which will effectively flip the sign of the function on one side of the root.

We then update $(x_mathrm L,x_mathrm M,x_mathrm R)$ so that $f(x_mathrm M)$ is minimal (between $f(x_mathrm M)$ and $f(x_mathrm N)$) and $x_mathrm L$ and $x_mathrm R$ are the nearest points found on the left and right sides of $x_mathrm M$.

See this graph for a visualization of the process. (There is a slight mistake in the conditions for updating, but the first 3 iterations are still correct.)

See here for code.