Robust formula for $N$-point Padé approximant calculation based on Wynn identity

TL;DR

The paper introduces a robust empirical criterion for selecting the optimal $N$-point Padé approximant by leveraging Wynn's identity, defining $C \equiv r_{l,m}$ and neighboring approximants to form a minimal-$|\eta|$ measure ($\eta_{\min}$) that also serves as an error proxy. It presents a concrete algorithm (the CNEWS framework) to construct Padé approximants via the Wynn relation, combined with Eitken interpolation to handle multipoint data, and demonstrates its effectiveness on divergent-series summation and sine-arch extrapolation, with accompanying code and generalization to derivatives at nodal points. The key contributions include a universal, practical selector for Padé approximants, a new N-point Padé computation formula, and empirical evidence that $\eta_{\min}$ tracks the real error, enabling reliable extrapolation and series summation in physics applications and beyond. These results have broad implications for solving differential equations, numerical analytic continuation, and predictor-corrector schemes, and suggest potential integration into standard numerical software.

Abstract

The performed numerical analysis reveals that Wynn's identity for the compass $1/(N-C)+1/(S-C)=1/(W-C)+1/(E-C)=1/η$ (here C stands for center, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal $|η|$, for the choice of the optimal Padé approximant. The work of this method is illustrated by calculation of multipoint Padé approximation by a new formula for calculation of this best rational approximation. The work of the criterion for the calculation of optimal Padé approximant is illustrated by the frequently seen in the theoretical physics problems of calculation of series summation and multipoint Padé approximation used as a predictor for solution of differential equations motivated by the magneto-hydrodynamic problem of heating of solar corona by Alvén waves. In such a way, an efficient and valuable control mechanism for $N$-point Padé approximant calculation is proposed. We believe that the suggested method and criterion can be useful for many applied problems in numerous areas not only in physics but in any scientific application where differential equations are solved. The solution of the Cauchy-Jacobi problem is illustrated by a Fortran program. The algorithm is generalized for the case of the first $K$ derivatives at $N$ nodal points.

Figures (6)

• Figure 1: The logarithmic function (line) and the series summation with $\eta_\mathrm{min}$ criterion for the optimal choice (dots). The calculated values are evaporated from the analytical curve, but $\eta_\mathrm{min}$ criterion gives reliable warning depicted in Fig. \ref{['fig:log101ee']}.
• Figure 2: Square of decimal logarithms of empirical $\varepsilon_\mathrm{emp} \equiv \eta_\mathrm{min}$ and real $\varepsilon_\mathrm{real}$ error versus the argument of the function for the calculation of the $\ln(1+x)$ series in Fig. \ref{['fig:log101']}. Close to the convergence radius the errors are small and almost linearly increase, then the errors reach saturation when the numerical resources of the fixed accuracy are exhausted.
• Figure 3: The logarithm of the empirical $\eta_\mathrm{min} \equiv \varepsilon_\mathrm{emp}$ versus logarithm of the real $\varepsilon_\mathrm{real}$ error for the calculation of the $\ln(1+x)$ series in Fig. \ref{['fig:log101']}. The high correlation coefficient 0.961 of the linear regression reveals that the long sought criterion for empirical evaluation of the accuracy of the Padé approximants calculated by $\varepsilon$-algorithm has already been found.
• Figure 4: Extrapolation of the function $\sin(x)$ (small dots) from 21 interpolation points (larger dots) in the interval $[-\pi,0]$ compared with the real function (line). In the interval $(0,\pi)$ the function is reliably extrapolated and the limit of the numerical implementation of the Aitken-Wynn extrapolation is clearly shown -- a gas of extrapolated points evaporated from the analytical function.
• Figure 5: Squared logarithm of the error estimates $\varepsilon$ of the $\sin(x)$ extrapolation shown in Fig. \ref{['fig:s21']}. The error $\varepsilon_\mathrm{real}$ is the real error of the extrapolation and the error $\varepsilon_\mathrm{emp} \equiv \eta_\mathrm{min}$. The similar behaviour of both errors shows that our criterion is a reliable method for error estimation, which is evident in Fig. \ref{['fig:s21e']}. The important problem in front of the applied mathematics is to research real extrapolation error beyond the extrapolation interval.