Convergence of a numerical method (${\max }_{t_n\le T}\Vert y_n-y(t_n)\Vert \to 0$)

For an initial value problem, convergence means that the largest global error on each fixed finite time interval tends to zero as the step size tends to zero and the starting errors vanish. For linear multistep methods, the Dahlquist equivalence theorem separates this requirement into consistency of a numerical method and zero-stability.

An iterative sequence converges superlinearly to its limit when its nonzero error norms satisfy the displayed ratio condition, or when it reaches the limit in finitely many steps. Quadratic convergence is a stronger local rate. In contrast, exact linear convergence has a positive asymptotic error ratio below one. A superlinear method need not have quadratic convergence.

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.