  Density matrix perturbation theory (DMPT) is known as a promising alternative
to the Rayleigh-Schr\"odinger perturbation theory, in which the sum-over-state
(SOS) is replaced by algorithms with perturbed density matrices as the input
variables. In this article, we formulate and discuss three types of DMPT, with
two of them based only on density matrices: the approach of Kussmann and
Ochsenfeld [J. Chem. Phys.127, 054103 (2007)] is reformulated via the Sylvester
equation, and the recursive DMPT of A.M.N. Niklasson and M. Challacombe [Phys.
Rev. Lett. 92, 193001 (2004)] is extended to the hole-particle canonical
purification (HPCP) from [J. Chem. Phys. 144, 091102 (2016)]. Comparison of the
computational performances shows that the aformentioned methods outperform the
standard SOS. The HPCP-DMPT demonstrates stable convergence profiles but at a
higher computational cost when compared to the original recursive polynomial
method
