Modified Newton's Method multiplicity

Asked Mar 06 '16 at 15:05

Active Mar 06 '16 at 15:05

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I have an assignment question that I have no idea how to solve. It goes like this:

The multiplicity of the zero $x^*$ is the least integer $m$ such that $f^{(k)}(x^*)=0$ for $0 \le k \lt m$, but $f^{(m)}(x^*) \neq 0$. Show analytically that in the case of a zero of multiplicity $m$, the modified Newton's Method $$x_{n+1}=x_n-m \frac{f(x_n)}{f'(x_n)}$$ is quadratically convergent.

I have no clue how to solve this. Any help would be much appreciated. Thank you!

asked Mar 06 '16 at 15:05

David_T

I'm not sure I understand the answer to that one though. – David_T Mar 06 '16 at 15:09
Possible duplicate as well of http://math.stackexchange.com/q/312819 – Jean Marie Mar 06 '16 at 15:15
It reduces the problem to one with a simple root. The other variant is to analyze the function $f(x)=(x-a)^mq(x)$ and its derivative to find the next best root approximation from these two data points. – Lutz Lehmann Mar 06 '16 at 15:17
Got it. Thanks! – David_T Mar 06 '16 at 15:21

Modified Newton's Method multiplicity

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