= Quadratic convergence
{title2=$|e_{n+1}|\le C|e_n|^2$}
An iteration converges at least quadratically to a limit if its errors eventually satisfy the displayed bound with a fixed finite $C>0$. If $Ce_N<1$, induction gives $|e_{N+k}|\le C^{-1}(C|e_N|)^{2^k}$. Exact asymptotic quadratic convergence has $|e_{n+1}|/|e_n|^2\to c\in(0,\infty)$. <Newton root-finding iteration> at a simple scalar root has this order when the second derivative at the root is nonzero; if that derivative vanishes, convergence can be faster.
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