Question 3) Fixed point iteration: Let $a>0$ be given. Show that for any starting value $x_{0}>0$ the fixed point iteration $$ x_{t+1}=\frac{x_{t}^{3}+3 a x_{t}}3 x_{t}^{2}+a}, \quad t=0,1,2, \ldots $$ converges monotonically to $z=\sqrt{a}$. How big is the local convergence order? To do this, proceed as follows: - Show that $\sqrt{a}$ is a fixed point. Show that the sequence $\left(x_{n} ight)_{n}$ is monotonic for every starting value $x_{0}>0$. - Determine all possible limit values. Consider the expression $x_{t+1}-v$ a for the order of convergence CS.SD. 158