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The multipole moments are displayed in Tab.\ 1. They grow relatively fast with .
In Tabs. 2-5 we display for various combinations of r and the partial sums
and the transformed values
with , and corresponding to the recursion of the Legendre polynomials . These values are rounded and can not display more than 16 exact decimal digits. The definition Eq. (37) of the transformation is used. Also displayed is for both sequences and the number of exact digits. This is defined as the negative decadic logarithm of the relative error as indicated also in the table headers.
Also, we display in Figures 1, 2, and 3 graphically the performance of the acceleration method for the multipole expansion using partial sums up to . Plotted are the achievable number of exact digits without (Fig. 1) and with acceleration (Fig. 2), and the gain, i.e., the additional digits achieved using the acceleration method, as a function of r and . For simplicity, the -fold frequency approach was not used, and hence the performance is better for larger distances from the singularity at r=2 and x=1.
From these tables and figures, one observes a clear increase of the accuracy by using the extrapolation method. Already for smaller values of , there is a drastic reduction of the error. A gain of three and more digits is typical and corresponds to a reduction of the error by a factor of 1000 or more.
Let us remark that the additional numerical effort for the extrapolation is very low in comparison to the evaluation of the multipole moments.