Rates of convergence numerical analysis

Question

I have been leaning about fixed point iterations, and have been introduced to the notion of rates of convergence, in the quadratic, and linear case. Consider a fixed point iteration $x_{n+1} = g(x_n)$, then the iteration converges linearly if $|x_{n+1} - \alpha| \leq C|x_n - \alpha|$, $C > 0$, and quadratically if $|x_{n+1} - \alpha| \leq K|x_n - \alpha|^2$, $K > 0$.

Are there any different rates of convergence? What are examples of say, sequences that converge, but neither quadratically nor linearly?

Thank you.

Answer 1

In addition to my comments you may want to have a look at http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture03.pdf and https://en.wikipedia.org/wiki/Rate_of_convergence

Answer 2

Try $x_n := x^{3^n}$ with $0<x<1$ converging cubicly to zero.