On the Relationship between the Convergence Rates of Iterative and Continuous Processes

Considering iterative sequences that arise when approximate solutions $x_k$ to a numerical problem are updated by $x_{k+1} = x_k+v(x_k)$, where $v$ is a differentiable vector field, we derive necessary and sufficient conditions for such discrete processes to converge to a stationary point of $v$ at different Q-rates in terms of a similar notion of fast convergence for the corresponding continuous processes.