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This question already has an answer here:

I want list of failure cases for Newton-Raphson method. If possible please provide flow chart for Newton-Raphson method.

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marked as duplicate by Hans Lundmark, Jyrki Lahtonen May 16 '18 at 5:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Where the derivative is $0$ or undefined is an obvious case $\endgroup$ – Rhys Hughes May 15 '18 at 9:11
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    $\begingroup$ Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. $\endgroup$ – José Carlos Santos May 15 '18 at 9:12
  • $\begingroup$ i know few, i asked as it would be great if there will be a list of all possible failure cases at one place $\endgroup$ – abcdmath May 15 '18 at 9:29
  • $\begingroup$ If the function is too fast-changing / "noisy" and our derivative estimate does not take this noise into account. $\endgroup$ – mathreadler May 15 '18 at 9:33
  • $\begingroup$ With initial approximation $x_0$ to solution $x,$ if the derivative $ f' $ of the function $f$ vanishes or takes values arbitrarily close to $0$ on the open interval between $x_0$ and $ x$ then for some $n$ we may have $f'(x_n)=0$ for some $n$ or it may be that $f(x_n)\ne 0$ and that $f'(x_n) $ is so close to $0$ that $x_{n+1}$ can be "almost anywhere". $\endgroup$ – DanielWainfleet May 15 '18 at 13:23
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The Newton-Raphson Method is:

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

Since a fraction $\frac ab$ is undefined when $b=0$ or undefined, in this case that constiutes an equation $f(x)$ whose derivate cannot be found or is $0$.

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