  In this paper we study the impact of two types of preconditioning on the
numerical solution of large sparse augmented linear systems. The first
preconditioning matrix is the lower triangular part whereas the second is the
product of the lower triangular part with the upper triangular part of the
augmented system's coefficient matrix. For the first preconditioning matrix we
form the Generalized Modified Extrapolated Successive Overrelaxation (GMESOR)
method, whereas the second preconditioning matrix yields the Generalized
Modified Preconditioned Simultaneous Displacement (GMPSD) method, which is an
extrapolated form of the Symmetric Successive Overrelaxation method. We find
sufficient conditions for each aforementioned iterative method to converge. In
addition, we develop a geometric approach, for determining the optimum values
of their parameters and corresponding spectral radii. It is shown that both
iterative methods studied (GMESOR and GMPSD) attain the same rate of
convergence. Numerical results confirm our theoretical expectations.
