On the relationship between convergence rates of discrete and continuous dynamical systems

Considering iterative sequences that arise when the approximate solution to a numerical problem is updated by evaluating a vector field at the current iterate, we derive necessary and sufficient conditions for this procedure to converge to a stationary point of the vector field at different Q-rates. These conditions are characterised in terms of the differential properties of the vector field and the asymptotic dynamical behaviour of the associated continuous dynamical system.\ud \ud Raphael Hauser was supported through grant GR/S34472 from the Engineering and Physical Sciences Research Council of the UK.\ud \ud Jelena Nedic was supported through the Clarendon Fund, Oxford University Press and ORS award, Universities UK