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6.1 Multipole Expansion

The multipole moments tex2html_wrap_inline2358 are displayed in Tab.\ 1. They grow relatively fast with tex2html_wrap_inline2470 .

In Tabs. 2-5 we display for various combinations of r and tex2html_wrap_inline2520 the partial sums


and the transformed values


with tex2html_wrap_inline2840 , tex2html_wrap_inline2842 and tex2html_wrap_inline2844 corresponding to the recursion of the Legendre polynomials tex2html_wrap_inline2846 . These values are rounded and can not display more than 16 exact decimal digits. The definition Eq. (37) of the tex2html_wrap_inline2584 transformation is used. Also displayed is for both sequences tex2html_wrap_inline2850 and tex2html_wrap_inline2852 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 tex2html_wrap_inline2850 up to tex2html_wrap_inline2856 . 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 tex2html_wrap_inline2860 . For simplicity, the tex2html_wrap_inline2776 -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 tex2html_wrap_inline2470 , 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.

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Herbert H. H. Homeier (