Computational uncertainty principle in nonlinear ordinary differential equations - II. Theoretical analysis

The error propagation for general numerical method in ordinary differential equations ODEs is studied. Three kinds of convergence, theoretical, numeri cal and actual convergences, are presented. The various components of round -off error occurring in floating-point computation are fully detailed. By i ntroducing a new kind of recurrent inequality, the classical error bounds f or linear multistep methods are essentially improved, and joining probabili stic theory the "normal" growth of accumulated 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 phenom ena found in the numerical experiments in part I of this paper and derive t wo universal relations which are independent of types of ODEs, initial valu es and numerical schemes and are consistent with the numerical results. Fur thermore, we give the explicitly mathematical expression of the computation al uncertainty principle and expound the intrinsic relation between two unc ertainties which result from the inaccuracies of numerical method and calcu lating machine.