The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed.

翻译：递归 Neville 算法可用于递归计算插值函数。通过审慎选择用于插值（及外推）的横坐标，该算法可转化为一种加速收敛的方法。例如，可利用 Neville 算法从给定慢收敛输入级数的部分和中逐次消去求和上限的负幂项。本文证明，对于特定的外推横坐标选择，可将递归 Neville 格式替换为简单的一步变换，同时还能获取加速收敛后变换级数的次领头项。基于矩阵的统一公式可估计输入级数部分和趋于极限的收敛速率。作为应用实例，本文计算了氢原子 Bethe 对数至 100 位小数精度。此外，还讨论了该方法在余项可展开为逆阶乘级数或半整数幂级数情形下的推广。