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
\begin{equation} f(x) = 0 \Leftrightarrow x = \phi(x) := x + g(x)f(x) \end{equation}
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?