I have a problem with this exercise: Given the function
$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ $f\binom{x}{y}=\frac{1}{8}\binom{\ln(1+x^2+y^2)-1}{x^2+y^2}$
and the iteration:
$z_{k+1}=f(z_{k}) \text{ and } z_{0}=\binom{0}{0}$
Let $I := [−1, 1] ×[−1, 1]$ and $z^{*}$ the fixed point. I have to calculate how many iterations are needed to have an estimated error of
$\left \|z_{k}-z^{*} \right \|_{\infty}\leq 10^{-3}$
How can I start this problem? Is there for example a formula to solve it?
(IDK if it is important but I have alredy proved that the function converges to a fixed point)
Hint
For $k \ge 1$, you have using the Mean Value Theorem, and the fact that $z^* = f(z_*)$
$$\begin{aligned} \lVert z_{k+1} - z^* \rVert &= \lVert f(z_k) - f(z^*)\rVert\\ &\le M \lVert z_{k} - z^* \rVert\\ &\le M^{k+1} \lVert z_{0} - z^* \rVert\\ &\le \sqrt 2 M^{k+1} \end{aligned}$$ where $M = \sup\limits_{x \in I} \lVert J_f(x) \rVert$ and $J_f$ the Jacobian of $f$.
Hence if you bound $M$, you can estimate $k$ such that $\sqrt 2 M^{k+1} \le 10^{-3}$.