Performance of superconvergent perturbation theory

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 beta, gives in the case of small coupling constants somewhat better results than the strongly divergent but asymptotic Rayleigh-Schrodinger perturbation series if it is truncated at the same order in beta. However, the transformation of this truncated perturbation series into Pade approximants or into another class of rational functions by means of the sequence transformation delta(k)((n))(zeta,s(n)) [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 delta(k)((n))(zeta,s(n)). Finally, it is shown that the other rational approximants provide much better approximations to higher order terms of the Rayleigh-Schrodinger perturbation series than Scherer's superconvergent approximation. [S1050-2947(97)02112-4].