  We performed comparative analysis for DFT+$U$ functionals to better
understand their applicability to non-collinear magnetism. Taking LiNiPO$_4$
and Sr$_2$IrO$_4$ as examples, we investigated the results out of two
formalisms based on charge-only density and spin density functional plus $U$
calculations. Our results show that the ground state spin order in terms of
tilting angle is strongly dependent on Hund $J$. In particular, the opposite
behavior of canting angles as a function of $J$ is found for LiNiPO$_4$. The
dependence on the other physical parameters such as Hubbard $U$ and Slater
parameterization $F^4/F^2$ is investigated. We also discuss the formal aspects
of these functional dependences as well as parameter dependences. The current
study provides useful information and important intuition for the
first-principles calculation of non-collinear magnetic materials.
