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In Tabs. 6-8 we plot for various combinations of r and the partial sums
of the exact expansion (45) and the transformed values
with , and corresponding to the recursion of the Legendre polynomials . As in the case of the multipole expansion the values are rounded to 16 decimal digits. The definition Eq. (37) of the transformation is used. Also, we plot for both sequences and the corresponding number of exact digits. As before, this number is defined as the negative decadic logarithm of the relative error as indicated also in the table headers.
Comparison of Tabs. 2 with 6, 3 with 7 and 4 with 8, that have been computed for the same point, respectively, reveals that the converged extrapolated values differ considerably. This means that in this way the difference can be evaluated.
Let us remark that also in this case the additional numerical effort for the extrapolation is very low in comparison to the evaluation of the coefficients of the orthogonal expansion.
In summary, it can be stated that the acceleration of the expansion in Legendre polynomials via the transformation leads to pronounced error reduction, as well in the case of the multipole expansion as well as in the case of the exact computation of the electrostatic potential. Put another way, for achieving a certain accuracy, considerably less multipole moments or expansion coefficients, respectively, are necessary if a problem adapted extrapolation method as the transformation is used.