# Newton method and the Banach fixed-point theorem

I try to combine the Newton method and the Banach fixed-point theorem but I have still some questions:

Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous.

Let $f: I \rightarrow \mathbb{R}$ and $g: I \rightarrow \mathbb{R} \setminus \{0\}$ functions.

Now I know, that

$$f(x) = 0 \Leftrightarrow x = \phi(x) := x + g(x)f(x)$$

and that $\xi$ is a root of $f$ when $\xi$ is a fixed-point of $\phi$.

Now I want to approximate the root of $f$ by using the iteration sequence $x_{n+1} = \phi(x_n)$.

Now to my last questions:

1) Why do I choose $g(x) = -\frac{1}{f'(x)}$ ? How can I interpret $g(x)$ geometrically?

2) How can I provide conditions to $f$ so that the iteration sequence $x_{n+1} = \phi(x_n)$ converges for each starting value $x_0 \in I$ to the root of $f$. And can I choose an small interval around the root of $f$ so that the iteration sequence always converges?

Can somebody help me?

• Anybody how can help me? I am still trying to answer my questions but I did not succeed. Oct 21, 2014 at 16:44