Enhanced and Generalized One-Step Neville Algorithm: Fractional Powers and Access to the Convergence Rate

TL;DR

This work develops an enhanced Neville algorithm for accelerating convergence of slowly convergent nonalternating series by exploiting asymptotic remainder structures. By recasting the interpolation problem into a matrix framework with abscissas $x_i=1/(i+1)$, the authors derive universal formulas that yield not only the series limit $s_\infty$ but also leading subleading coefficients $c_j$, enabling quantification of the convergence rate. The method outperforms classical techniques like Wynn's $\varepsilon$-algorithm and Aitken's $\Delta^2$ in tested scenarios and avoids sampling very high-order terms required by some alternative resummations. Generalizations to half-integer and fractional power remainder terms (and even inverse factorial series) demonstrate the framework's versatility, with high-precision Bethe logarithms and new analytic forms for subleading coefficients serving as compelling demonstrations of practical impact. Overall, the enhanced Neville approach provides a robust, interpretable tool for convergence acceleration and tail analysis in physics-relevant series.

Abstract

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.

Figures (2)

• Figure 1: We investigate the convergence of the enhanced Neville transformation (green curve) for the input series \ref{['example']}, as measured by the quantity $\chi(n)$ defined in Eq. \ref{['chi']}. The quantity $\chi(n)$ roughly decreases by unity with every iteration of the algorithm, indicating the gain of roughly one converged decimal with every order of the enhanced Neville transformation. The convergence of the enhanced Neville transformation is compared with the convergence of the Wynn epsilon algorithm (blue curve), of the Aitken $\Delta^2$ process (red curve), and of the series itself (black curve). (For interpretation of the colors in the figure, the reader is referred to the current web version. The curves labeled "Series", "Wynn", "Aitken" and "Enhanced Neville" are listed from top to bottom both in the figure legend as well as in the figure itself.)
• Figure 2: The figures illustrate the convergence of sequences $d_j(n)$ toward $c_j$ for $j = 1, 2, 3, 4$, according to Eq. \ref{['eq:sigmas']}. The cases $j = 1, 2, 3, 4$ are treated in Fig. (a), (b), (d), and (d), respectively. The first 150 $d_j(n)$ with $n = 0, \dots, 149$ are compared to the value $c_j = d_j(\infty)$, the latter being denoted by a black diamond. One notes that $c_1 = -1$ is integer-valued, while the first 50 decimals of the coefficients $c_2$, $c_3$ and $c_4$ are given in Eq. \ref{['eq:coeffs']}.