Computational uncertainty principle in nonlinear ordinary differential equations

摘要

The error propagation for general numerical method in ordinary differential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components of round-off error occurring in floating-point computation are fully detailed. By introducing a new kind of recurrent inequality, the classical error bounds for linear multi- Step methods are essentially improved, and joining probabilistic theory the “normal” growth of accumu- lated round-off error is derived. Moreover, a unified estimate for the total error of general method is given. On the basis of these results, we rationally interpret the various phenomena found in the numer- ical experiments in part I of this paper and derive two universal relations which are independent of types of ODEs, initial values and numerical schemes and are consistent with the numerical results. Further- more, we give the explicitly mathematical expression of the computational uncertainty principle and ex- pound the intrinsic relation between two uncertainties which result from the inaccuracies of numerical method and calculating machine.