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From what I understand, for Newton's Method to converge in the equation

$$g(x_n) = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}$$

(1) $g(x)$ and $g'(x)$ must be continuous around the root.

(2) The initial guess must fall within this interval.

(3) The condition $|g'(x)| \leq 1$ for all $x$ in the interval around the root must be true.

This answer Showing that Newton's method converges confirms my idea. I also found When is Newton's Method guaranteed to converge to a good solution (non-linear system)? which has a good explanation of Newton's Method, but not really what I was looking for.

I just need to know if the above mentioned conditions guarantee convergence?

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Richard Slabbert
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  • What is the interval you designate in (2) as "this interval" ? 2) No, these conditions are not necessary neither sufficient.
    • – Jean Marie Feb 26 '16 at 18:02
  • "The interval" is the interval around the root where $|g'(x)| \leq 1$. – Richard Slabbert Feb 26 '16 at 18:04
  • @JeanMarie So these conditions do not guarantee convergence? Are there conditions that do guarantee convergence? – Richard Slabbert Feb 26 '16 at 18:06
  • There are. But in general there is nothing that you can check a priori. – user251257 Feb 26 '16 at 18:11
  • There are such conditions as treated in the following paper you can fiind on the web vberinde.ubm.ro/.../ANUOC-1995-Berinde.PDF Usually they are very restrictive... – Jean Marie Feb 26 '16 at 18:15
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    The problem of determining where Newton's method converges can be truly difficult and even chaotically behaved, as illustrated in the paper "On the iteration of a rational function: computer experiments with Newton's method" by Curry, Garnett, and Sullivan. – Lee Mosher Feb 26 '16 at 18:22
  • I found this answer on this site: Sufficient conditions for the convergence of Newton's Method which is very similar to the conditions above. – Richard Slabbert Feb 26 '16 at 18:25