# How does the modified Newton's method for multiple roots converge quadratically?

I am currently learning about the multiple root issue for Newton's method. My textbook hinted at tweaking the function using $$h(x)=\frac{f(x)}{f′(x)}$$ and making the iteration function to actually be $$g(x)= x - \frac{h(x)}{h'(x)}$$

My textbook claims that this method is also quadratically convergent but provided no proof and I am lost as to how it is quadratically convergent. I tried to do some more research but all I could find is this: https://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/iteration%20methods/accelerating%20the%20convergence/mrac.html

Can anybody show me how this quadratically convergent or at least point me in the direction to prove it myself?

• If $r$ is the root of $f$, then $r$ is a root of $h$ with multiplicity $1$. Alternative: It should be possible to show that $g'(r) = 0$. This will also be enough, depending on what theorems you already have. Commented Jan 31, 2021 at 11:00
• I understand how if r is a root of f then r is a root of h with multiplicity 1 but I am failing to understand how that implies quadratic convergence. Could you elaborate a little? Commented Jan 31, 2021 at 14:13
• Local quadratic convergence is ensured when the the function is sufficiently smooth and the root has multiplicity $1$. Commented Jan 31, 2021 at 14:35
• There are some related posts on this [here]( math.stackexchange.com/…). It is likely that they can help you. Commented Jan 31, 2021 at 20:02

## 1 Answer

Hint: That method is known as Schröder's Method and the iteration formula may be obtained by applying Newton's method to $$f(x)/f'(x)$$ instead of $$f(x)$$.