0
$\begingroup$

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

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

$\endgroup$
1
$\begingroup$

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 $|\phi'(c)|<1$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.