Within the framework of perturbation theory we propose, firstly, an iterative method which may serve as a source of optimal unperturbed solutions in both one and more dimensions. It combines the Runge - Kutta and Newton algorithm and its efficiency is illustrated on a few quartic oscillators. Secondly, admitting also an arbitrary perturbation of potentials we generalize the existing Runge - Kutta one-dimensional Rayleigh - Schrödinger constructions of energies and wavefunctions to more dimensions.