  We consider the iterative solution of large linear systems of equations in
which the coefficient matrix is the sum of two terms, a sparse matrix $A$ and a
possibly dense, rank deficient matrix of the form $\gamma UU^T$, where $\gamma
> 0$ is a parameter which in some applications may be taken to be 1. The matrix
$A$ itself can be singular, but we assume that the symmetric part of $A$ is
positive semidefinite and that $A+\gamma UU^T$ is nonsingular. Linear systems
of this form arise frequently in fields like optimization, fluid mechanics,
computational statistics, and others. We investigate preconditioning strategies
based on an alternating splitting approach combined with the use of the
Sherman-Morrison-Woodbury matrix identity. The potential of the proposed
approach is demonstrated by means of numerical experiments on linear systems
from different application areas.
