Fixed Point Iteration and Order of Convergence of a function

Question

For the function f(x) = cos(cosx), does the fixed point iteration for finding the fixed point in [0,1] converge for all first points, p in [0,1]? If it does, what is the order of convergence? I am having trouble starting on this question, can anyone explain how to start, and how can I find the order of convergence?

Answer 1

If you observe that $f(\mathbb{R}) = [\cos 1, 1]=I$, you can see that if you establish convergence for any initial approximation $x_0 \in I$ you will have established convergence for any $x_0 \in \mathbb{R}$. Since $f$ is obviously invariant in $I$ and $|f'(x)| < \frac 12$ in $I$, the fixed point theorem guarantees that $f$ has a unique fixed point in $z\in I$ and that the fixed point iteration converges to $z$ for any $x_0 \in I$ (actually any $x_0\in \mathbb{R}$).

Regarding the order of convergence, it can only be greater than $1$ if $f'(z)=0$. However, this would only hold if $z = k \pi, k \in \mathbb{Z}$, which are not fixed points of $f$.