The error propagation for general numerical method in ordinarydifferential equations ODEs is studied. Three kinds of convergence,theoretical, numerical and actual convergences, are Presented. Thevarious components of round-off error occurring in floating-pointcomputation are fully Detailed. By introducing a new kind ofrecurrent inequality, the classical error bounds for linear multi- Step methods are essentially improved, and joining probabilistictheory the "normal" growth of accumu- lated round-off error isderived. Moreover, a unified estimate for the total error of generalmethod is Given. On the basis of these results, we rationallyinterprete the various phenomena found in the numer- ical experimentsin part l of this paper and derive two universal relations which areindependent of types Of ODEs, initial values and numerical schemesand are consistent with the numerical results. Further- More, we givethe explicitly mathematical expression of the computationaluncertainty principle and ex- Pound the intrinsic relation betweentwo uncertainties which result from the inaccuracies of numericalMethod and calculating machine.
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