I've read several references, and it is true that:
- A point is called a fixed point if $f(x_0) = x_0$.
- It can further be reduced to find root of a non-linear function $f(x) = g(x) - x = 0$
- The fixed point iteration method will converge when $|g'(r)| < 1$, for a root $r$.
- The fixed point iteration normally will have linear convergence rate, meaning the error will decrease linearly. $|e_{i+1}| = k|e_{i}|^p$, and $p = 1$.
- The fixed point iteration will converge at least quadratically when $g'(r) = 0$, $|e_{i+1}| = k|e_{i}|^p$, and $p \geq 2$.
Now, let's say I have a function $x^2 = \cos(\pi x)$. One of the fixed point iteration function $g(x)$ we can use is $x = g(x) = \frac{\arccos(x^2)}{\pi}$. But this of course, I have checked, will converged linearly.
My question is, how do you find a function $g(x)$ so that it will at least quadratically converge? (I don't want to use the newton-raphson's method). Is there any approach to find it that I miss? Thank you