  The existence of light sterile neutrinos is a long standing question for
particle physics. Several experimental ``anomalies'' could be explained by
introducing ~eV mass scaled light sterile neutrinos. Many experiments are
actively hunting for such light sterile neutrinos through neutrino oscillation.
For long baseline experiments, matter effect needs to be treated carefully for
precise neutrino oscillation probability calculation. However, it is usually
time-consuming or analytical complexity. In this manuscript we adopt the
Jacobi-like method to diagonalize the Hermitian Hamiltonian matrix and derive
analytically simplified neutrino oscillation probabilities for 3 (active) + 1
(sterile)-neutrino mixing for a constant matter density. These approximations
can reach quite high numerical accuracy while keeping its analytical simplicity
and fast computing speed. It would be useful for the current and future long
baseline neutrino oscillation experiments.
