  With the continuous accumulation of data from hadron collision experiments,
efficient Partial Wave Analysis tools are indispensable for constructing a
clear hadron spectrum. Currently, automated computations of scattering
amplitudes primarily use the helicity scheme and covariant effective Lagrangian
method. The automated calculations under the covariant $L$-$S$ scheme, which is
one of the commonly used partial wave analysis schemes, have not been fully
realized. In this work, we provide a general algorithm for computing partial
wave amplitudes under the covariant $L$-$S$ scheme. This will lay the
foundation for automated computation of partial wave amplitudes under the
covariant $L$-$S$ scheme.
