TITLE: Relationship between Newton's method an fixed-point iteration
QUESTION [5 upvotes]: The formula for Newton's iteration method (which is a zero-finding problem) is $$ x_{k+1}=x_{k} - \frac{f(x_k)}{f'(x_k)}$$
I read in my textbook that this can be also be seen as a fixed-point iteration, where the zero of this function $f$ is a fixed point for another function $g$. Tha is, something like this: $$g(x)=x - \frac{f(x)}{f'(x)}$$ 
My question is, why is this useful? 
For example if we wanted to prove that a function $f(x)=x^2-2$ converges in the interval $[1,2]$, how could the fixed point property be useful? I don't see what the point is, and how it is different than just using Newton's method.

REPLY [2 votes]: Fixed point iteration means that $x_{n+1}=f(x_n)$
Newton's Method is a special case of fixed point iteration for a function $g(x)$ where $x_{n+1}=x_n-\frac{g(x_n)}{g'(x_n)}$
If you take $f(x)=x-\frac{g(x)}{g'(x)}$ then Newton's Method IS indeed a special case of fixed point iteration.
This means that everything that you know about fixed point iteration also applies to Newton's Method; which is useful, since much is know about the behaviour of fixed point iteration.