# Starting solution of a Newton Method

I was going through some computations and I found something strange about Newton's Method. If anyone can verify it, it would be of great help.

Newton's method is given by $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. Now, say I have a non differentiable function $f(x)$. But, the point of non- differentiabilty is away from the zero. Hence, if I apply Newton's Method with starting solution close to the root, in some cases I will get the solution (as desired).

Now, I think for smooth functions the starting value don't play an important role in finding the root. It surely helps in reducing the number of iterations but even if we start from some not so close point, we will still get the answer (Yes, the number of iterations will increase).

Whereas, for Non-Smooth functions the starting solution should be close to zero (again provided the non differentiable point is away from the root). If we start from some far away point then because of non differentiability we will have issues with the method.

Is my thinking, correct?

You exclude the possibility that the point of nondifferentiability is at the root, but this can be overoptimistic. For instance, the function $\sqrt[3]{x}$ has a root at $x=0$ and is not differentiable at $x=0$. In this example, iteration doubles the distance from the origin.