  We modify a non-perturbative approach based on the Principle of Minimal
Sensitivity to calculate the periods of relativistic oscillators with even
polynomial potentials of general form. The optimization of the variational
parameter is adapted by using the ultrarelativistic limit of the period as a
boundary condition, which determines the values of additional free parameters
introduced. Resulting general approximations prove to be accurate over the
whole solution domain and for the whole class of potentials.
