  We extend a simple and compact method for calculating the three flavor
neutrino oscillation probabilities in uniform matter density to schemes with
sterile neutrinos, with favorable features inherited. The only constraint of
the extended method is that the scale of the matter potential is not
significantly larger than the atmospheric $\Delta m^2$, which is satisfied by
all the running and proposed accelerator oscillation experiments. Degeneracy of
the zeroth order eigensystem around solar and atmospheric resonances are
resolved. Corrections to the zeroth order results are restricted to no larger
than the ratio of the solar to the atmospheric $\Delta m^2$. The zeroth order
expressions are exact in vacuum because all the higher order corrections vanish
when the matter potential is set zero. Also because all the corrections are
continuous functions of matter potential, the zeroth order precision is much
better than $\Delta m^2_\odot/\Delta m^2_\text{atm}$ for weak matter effect.
Numerical tests are presented to verify the theoretical predictions of the
exceptional features. Moreover, possible applications of the method in
experiments to check the existence of sterile neutrinos are discussed.

High Energy Physics - Phenomenology