This paper proposes a method based on the compressed-map principle for determination of the initial powers of the small parameter when the solutions are expanded in asymptotic series. To find a given solution, an initial approximation is chosen and simple iteration is used to calculate weighting coefficients in terms of the small parameters for each unknown in the system equation. The differentiation symbols are endowed with algebraic sense by representing them with equation coefficients expressed in terms of the small parameters. The weighting coefficients found as a result of the first iteration are the initial powers in the asymptotic expansions of the unknowns. Asymptotic convergence and convergence of the simple iteration procedure follows from the existence conditions for the asymptotic expansions. Depending on the choice of initial approximation, the first iteration results in more or less familiar theories (nonmoment, seminonmoment) of the simple edge effect in shallow shells and shells of negative curvature. Estimates of their accuracy are naturally given at this point. Thus, it can be inferred that the compressed-image principle separates simpler subtheories from the general theory and that these subtheories are in turn theories of elementary states of stress, i.e., possess decomposition properties.