  Iterative methods based on matrix splittings are useful in solving large
sparse linear systems. In this direction, proper splittings and its several
extensions are used to deal with singular and rectangular linear systems. In
this article, we introduce a new iteration scheme called three-step alternating
iterations using proper splittings and group inverses to find an approximate
solution of singular linear systems, iteratively. A preconditioned alternating
iterative scheme is also proposed to relax some sufficient conditions and to
obtain faster convergence as well. We then show that our scheme converges
faster than the existing one. The theoretical findings are then validated
numerically.
