On the relationship between the convergence rates of iterative and continuous processes

Considering iterative sequences that arise when the approximate solution x k to a numerical problem is updated by x k+1 = x k + v(x k), where v is a vector field, we derive necessary and sufficient conditions for such discrete methods to converge to a stationary point of v at different Q-rates in terms of the dierential properties of v and in terms of the asymptotic dynamical behaviour of the associated continuous dynamical system