I have been asked if the Newton's method (More concretely Maehly's method) is numerically inestable. I know finding roots of a polynomial is an ill-posed problem, and also that the Newton's method converges to a root $x$ in a small enough neighborhood of $x$.
What can I say about stability? I would say that Newton's method is indeed numerically stable provided the polynomial does not have roots too close to each other, because in practice when two or more roots are too close from each other in the iterations
$$x_{k+1}=x_k-\frac{p(x_k)}{p'(x_k)-p(x_k) \sum_{j=1}^m \frac{1}{x - \alpha_j} }, $$ where the $\alpha_j$ are the already computed roots, you are substracting two terms that are very close, $p'(x_k)-p(x_k)$ (catastrophic cancelation), and then also dividing by a number close to $0$, because both terms tend to $0$.
What do you think? Is this answer correct?