  An improved preconditioned conjugate gradient squared (PCGS) algorithm has
recently been proposed, and it performs much better than the conventional PCGS
algorithm. In this paper, the improved PCGS algorithm is verified as a
coordinative to the left-preconditioned system, and it has the advantages of
both the conventional and the left-PCGS; this is done by comparing, analyzing,
and executing numerical examinations of various PCGS algorithms, including
another improved one. We show that the direction of the preconditioned system
for the CGS method is determined by the operations of $\alpha_k$ and $\beta_k$
in the PCGS algorithm. By comparing the logical structures of these algorithms,
we show that the direction of the preconditioned system can be switched by the
construction and setting of the initial shadow residual vector.
