Newton iteration at a multiple root ($e_{k+1}/e_k\to (m-1)/m$)

If a sufficiently smooth function has the form $f(x)=(x-\alpha {)}^{m}q(x)$ with $m>1$ and $q(\alpha )\ne 0$, then the Newton root-finding iteration satisfiesThus the ratio of successive nonzero errors tends to $(m-1)/m$: the usual iteration has only linear convergence. Cancelling the common factor in $f/f^{\prime }$ produces a continuous extension at the multiple root, although the original quotient is undefined there. An implementation should stop upon finding an exact zero of a function.