  The three-step alternating iteration scheme for finding an iterative solution
of a singular (non-singular) linear systems in a faster way was introduced by
Nandi {\it et al.} [Numer. Algorithms; 84 (2) (2020) 457-483], recently. The
authors then provided its convergence criteria for a class of matrix splitting
called proper G-weak regular splittings of type I. In this note, we analyze
further the convergence criteria of the same scheme. In this aspect, we obtain
sufficient conditions for the convergence of the same scheme for another class
of matrix splittings called proper G-weak regular splittings of type II. We
then show that this scheme converges faster than the two-step alternating and
usual iteration schemes, even for this class of splittings. As a particular
case, we also establish faster convergence criteria of three-step in a
nonsingular matrix setting. This is shown that a large amount of computational
time and memory are required in single-step and two-step alternating iterative
methods to solve the nonsingular linear systems more efficiently than the
three-step alternating iteration method. Finally, the semiconvergence of a
three-step alternating iterative scheme is established. Its faster
semiconvergence is demonstrated by considering a singular linear system arising
from the Markov process.
