[Math] how to prove the convergence of fixed point iteration algorithm

absolute-convergencealgorithmsconvergence-divergencerecursive algorithms

Please refer to the below algorithm:

enter image description here

Above two steps can be rewritten as,
\begin{equation}
x(k+1)=\arccos\bigg( -\frac{1}{2(Dr^{\frac {|\sin(2x(k)+\theta)|}{M\sin x(k)\sqrt{A+2B\cos(2x(k)+\theta)}}}+1)} \bigg) \nonumber
\end{equation}

\begin{equation}
x_{k+1}=f(x_{k})
\end{equation}

The work I did was above equations can be written as $x_{k+1}=f(x_{k})$ If,

1) $f'(n) \neq 0$, then the sequence converges linearly to the fixed point $n$.

2) $f'(n) = 0$, then the sequence converges at least quadratically to the fixed point $n$.

But unfortunately first derivative is a messy equation and hard to prove

so any idea to prove the convergence of the algorithm?

Best Answer

Do you know which quadrant for $\arccos$?

Clearly the iterative function maps $[-\pi,\pi]$ into itself. So you should be able to establish the existence of the fixed point by Brouwer's fixed point theorem. Beyond that you need some proof of contraction. Your formula is a mess, so no idea how you will do it.

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