Performance of superconvergent perturbation theory

Details Performance of superconvergent perturbation theory Performance of superconvergent perturbation theory

Dec 1997

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1997phrva..56.5165w&link_type=abstract

Physical Review A (Atomic, Molecular, and Optical Physics), Volume 56, Issue 6, December 1997, pp.5165-5168

Statistics

Computation

19

Quantum Mechanics, Sequences, Series, And Summability, Computational Techniques, Simulations

Scientific paper

The performance of the so-called superconvergent perturbation theory [W. Scherer, Phys. Rev. Lett. 74, 1495 (1995)] is investigated numerically in the case of the ground-state energy of a quartic anharmonic oscillator. It is shown that Scherer's superconvergent approximation, which is rational in the coupling constant β, gives in the case of small coupling constants somewhat better results than the strongly divergent but asymptotic Rayleigh-Schrödinger perturbation series if it is truncated at the same order in β. However, the transformation of this truncated perturbation series into Padé approximants or into another class of rational functions by means of the sequence transformation δ(n)k(ζ,sn) [E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989)] yields much more powerful rational approximants. Moreover, the performance of the superconvergent approximation can be improved considerably by Wynn's epsilon algorithm [P. Wynn, Math. Tables Aids Comput. 10, 91 (1956)] or by δ(n)k(ζ,sn). Finally, it is shown that the other rational approximants provide much better approximations to higher order terms of the Rayleigh-Schrödinger perturbation series than Scherer's superconvergent approximation.

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