Let's say I have a function with with a root a of order m. I know I can write my function, $f(x)$, as $f(x) = (x-a)^mh(x)$, where h(x) doesn't have a zero in a neighborhood of a. I want to prove that the NR method converges linearly if I make the iterations as $g(x) = x - \frac{f(x)}{f'(x)}$ and quadratically if $g(x) = x - m\frac{f(x)}{f'(x)}$. If I plug in $f(x)$ to the equation and derive I don't get anything I'm looking for. I feel like I'm doing something wrong. Any help would be appreciated!
2026-03-28 10:15:03.1774692903
Newton-Raphson iterative method when derivative is $0$?
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Clearly the method doesn't work in these cases. Pick a different starting point and then iterate. If you are in the appropriate attraction basin, yo will find the point you are looking for. To understand this geometrically, realize that points with horizontal tangent lines will never intersect the x axis, thus not yielding a new point to continue iterating.