# Show that the newton-rhapson iteration is a contraction under certain conditions

The Newton–Raphson method for finding roots of a differentiable function $f : \mathbb{R} \to \mathbb{R}$ is to iterate the function $$x \mapsto g(x):=x - \frac{f(x)}{f'(x)}$$ Where the following conditions is assumed on $f$

1. $f$ has a continous second derivative
2. $f'(x) \neq 0 \ \forall \ x \in \mathbb{R}$
3. There exists some $\alpha \in (0,1)$ such that $\left| f(x) f''(x)\right| \, \leq \, \alpha \left| f'(x) \right|^2 \ \forall \ x \in \mathbb{R}$

The question is how to prove that $g$ is a contraction.

I have already tried to use the definition of a contraction, that there exists some $0<\alpha<1$ such that

$$d(Tx,Ty) \leq \alpha d(x,y),$$

holds. Now I tried using the metric induced by the supremum norm. But alas this gave me nothing. The hint was to use the Mean Value Theorem, but I can not quite see how that applies here. Can someone show me/ give me some clear suggestions on how to proceed? =)

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Is this homework? If so please tag it as such. – Epictetus Dec 3 '12 at 16:21
No, preparing for an exam.. This was one of the last questions. See here for a source math.ntnu.no/~stacey/documents/tma4145/finalH09en.pdf. – N3buchadnezzar Dec 3 '12 at 16:23
Hint: What is $g'(x)$? – Thomas Andrews Dec 3 '12 at 16:28
Why are you using the "supremum norm" metric, when your space is just $\mathbb R$? You are trying to show $g(x)$ is a contraction map, and $g(x):\mathbb R\rightarrow\mathbb R$. – Thomas Andrews Dec 3 '12 at 16:32

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