增强型广义单步Neville算法：分数幂与收敛速度的获取
Enhanced and Generalized One-Step Neville Algorithm: Fractional Powers and Access to the Convergence Rate
AI总结：
本研究提出一种基于特定横坐标选择的增强型广义单步Neville算法，替代递归格式实现收敛加速并获取次领头项，可估计收敛速度，将氢原子Bethe对数算至100位小数，还可推广至逆阶乘、半整数幂级数。
摘要：
递归Neville算法可用于递归计算插值函数。通过明智地选择插值（及外插）所用的横坐标，该算法可衍生出一种收敛加速方法。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逆幂次。本文表明，对于外插所用横坐标的一种特定选择，可将递归Neville格式替换为简单的单步变换，同时还能获取收敛加速后变换级数的次领头项。基于矩阵的统一公式可用于估计输入级数的部分和趋近其极限的收敛速度。具体而言，氢原子的Bethe对数被计算至100位小数。本文还讨论了该方法的推广情形，适用于余项可展开为逆阶乘级数的级数，或带有半整数次幂的级数。
英文摘要：
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.
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