Self-similar perturbation theory

A method is suggested for treating those complicated physical problems for which exact solutions are not known but a few approximation terms of a calc ulational algorithm can be derived. The method permits one to answer the fo llowing rather delicate questions: What can be said about the convergence o f the calculational procedure when only a few of its terms are available an d how to decide which of the initial approximations of the perturbative alg orithm is better, when several such initial approximations are possible? De finite answers to these important questions become possible by employing th e self-similar perturbation theory. The novelty of this paper is in develop ing the stability analysis based on the method of multipliers and in illust rating the efficiency of this analysis by different quantum-mechanical prob lems. (C) 1999 Academic Press.