  Saddle-point systems appear in many scientific and engineering applications.
The systems can be sparse, symmetric or nonsymmetric, and possibly singular. In
many of these applications, the number of constraints is relatively small
compared to the number of unknowns. The traditional null-space method is
inefficient for these systems, because it is expensive to find the null space
explicitly. Some alternatives, notably constraint-preconditioned or projected
Krylov methods, are relatively efficient, but they can suffer from numerical
instability and even nonconvergence. In addition, most existing methods require
the system to be nonsingular or be reducible to a nonsingular system. In this
paper, we propose a new method, called OPINS, for singular and nonsingular
saddle-point systems. OPINS is equivalent to the null-space method with an
orthogonal projector, without forming the orthogonal basis of the null space
explicitly. OPINS can not only solve for the unique solution for nonsingular
saddle-point problems, but also find the minimum-norm solution in terms of the
solution variables for singular systems. The method is efficient and easy to
implement using existing Krylov solvers for singular systems. At the same time,
it is more stable than the other alternatives, such as projected Krylov
methods. We present some preconditioners to accelerate the convergence of OPINS
for nonsingular systems, and compare OPINS against some present
state-of-the-art methods for various types of singular and nonsingular systems.
