Cubic convergence
= Cubic convergence
{title2=$\|e_{k+1}\|\leq C\|e_k\|^3$}
An iteration has at least cubic convergence when its errors eventually satisfy the displayed inequality with fixed $C>0$. Exact <order of convergence> three requires a positive finite limit of $\|e_{k+1}\|/\|e_k\|^3$. If $C^{1/2}\|e_N\|<1$, setting $d_k=C^{1/2}\|e_k\|$ gives $d_{k+1}\leq d_k^3$, hence $d_{N+j}\leq d_N^{3^j}$.