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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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This content will become publicly available on November 1, 2026
Global rational approximations of functions with factorially divergent asymptotic series
We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to . We show that dyadic expansions are numerically efficient representations. For special functions such as Bessel, Airy, Ei, erfc, Gamma, etc. the region of convergence of dyadic series is the complex plane minus a ray, with this cut chosen at will. Dyadic expansions thus provide uniform, geometrically convergent asymptotic expansions including near antistokes rays. We prove that relatively general functions, Écalle resurgent ones, possess convergent dyadic expansions. These expansions extend to operators, resulting in representations of the resolvent of self-adjoint operators as series in terms of the associated unitary evolution operator evaluated at some prescribed discrete times (alternatively, for positive operators, in terms of the generated semigroup).
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- Award ID(s):
- 2206241
- PAR ID:
- 10592491
- Publisher / Repository:
- Journal of Approximation Theory
- Date Published:
- Journal Name:
- Journal of Approximation Theory
- Volume:
- 311
- Issue:
- C
- ISSN:
- 0021-9045
- Page Range / eLocation ID:
- 106178
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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