  In this paper, a class of new preconditioners based on matrix splitting are
presented for generalized saddle-point linear systems, which can be viewed as
further modified improvements of some recently published preconditioners.
Moreover, we widen the scope of the new preconditioners to solve the more
general but rarely considered saddle-point linear systems with singular leading
blocks and rank-deficient off-diagonal blocks. The new variants can result in
much better convergence properties and spectrum distributions than the original
existing preconditioners. Numerical experiments are used to illustrate the
efficiency of the new proposed preconditioners.
