Question

3. Find all fixed points for the associated Newton iteration function for F(a) z/ 1) when n-1,2,3... Which are attracting and which are repelling?

Answer 1

Let us write

F,,(x) = (z-1)n

The associated Newton iteration function is

Fn(a) Fn (x) G,(r) = r- (r-1)n r(r 1) n.r2

The fixed points are given by

nr2 2

The solutions to $\begin{align*}x^2-x&=0\end{align*}$ are $\begin{align*}x=0,1\end{align*}$ . Hence, these are the only fixed points.

Since

n.a2

we get

(z) =-

we have

$\begin{align*}G_n'(0)&=0\\ G_n'(1)&={\frac{n+n^2}{n^2}}\end{align*}$

Since $\begin{align*}|G_n'(0)|&=0\end{align*}$ , the fixed point $\begin{align*}x=0\end{align*}$ is attracting; since

n+n

the fixed point $\begin{align*}x=1\end{align*}$ is repelling.