# Modified Newton method - explanation

Could someone describe the Modified Newton method or give a reference?

Method: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_0)}$$

The modified method (but also the original method) can be seen as a special case for fixpoint iteration. That is to solve the fixpoint problem $\phi(x)=x$ by iteratively apply $\phi$ to approximations (ie $x_{n+1} = \phi(x)$). This method converges at least if $|\phi'(x)|<1$ (in an neigborhood of the solution where we start).
Let $\phi(x) = x-f(x)/D$ then a method to find fixpoints to $\phi$ that is solve the equationn $\phi(x)=x$ (which is true iff $f(x)=0$) is to iteratively apply $\phi$. That is $x_{n+1} = \phi(x_n)$.
This method works if $|\phi'(c)|<1$ when $\phi(c)=c$, but $\phi'(x) = 1-f'(x)/D$. In order to ensure $|\phi'(c)|<1$ we select $D$ suitably. It's reasonable to assume that $f'(c)$ is approximately $f'(x_0)$ (if we select $x_0$ close to $c$) this means that setting $D=f'(x_0)$ will be a good candidate to make $|\phi'(c)|<1$