have been performed with very small errors. The averaged relative errors of total energy (∼10−9) and of total angular momentum (∼10−11) have remained nearly constant throughout the integration period (Fig. 1). The special startup procedure， warm start， would have reduced the averaged relative error in total energy by about one order of magnitude or more.
Relative numerical error of the total angular momentum δA/A0 and the total energy δE/E0 in our numerical integrationsN± 1，2，3， where δE and δA are the absolute change of the total energy and total angular momentum， respectively， andE0andA0are their initial values. The horizontal unit is Gyr.
Note that different operating systems， different mathematical libraries， and different hardware architectures result in different numerical errors， through the variations in round-off error handling and numerical algorithms. In the upper panel of Fig. 1， we can recognize this situation in the secular numerical error in the total angul
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