Quadratic convergence ($|e_{n+1}|\le C|e_n{|}^2$)

An iteration converges at least quadratically to a limit if its errors eventually satisfy the displayed bound with a fixed finite $C>0$. If $C{e}_N<1$, 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.