Why is the success of fixed point iteration determined by the the derivative of the function evaluated at the point of intersection?

Question

I can't seem to understand why cobwebs converge when the derivative of the function evaluated at the point of intersection is between 0 and -1, but cobwebs diverge is the derivative of the function is less than -1.

An intuitive explanation or proof would be appreciated.

Answer 1

Roughly speaking, the distance to the fixed point is multiplied by the derivative of the function at each step. Let the limit be $L$, so $L=f(L)$. If you write $x_{i+1}=f(x_i)$ and define $e_i=x_i-L$ we have $e_{i+1}=f(x_i)-L=f(L+e_i)-L\approx f(L)+e_if'(L)-L=e_if'(L)$ As long as $|f'(L)| \lt 1$ this will get each estimate closer to the limit. This shows you want the derivative between $-1$ and $1$.