  In this contribution, we study the numerical behavior of the Generalized
Minimal Residual (GMRES) method for solving singular linear systems. It is
known that GMRES determines a least squares solution without breakdown if the
coefficient matrix is range-symmetric (EP), or if its range and nullspace are
disjoint (GP) and the system is consistent. We show that the accuracy of GMRES
iterates may deteriorate in practice due to three distinct factors: (i) the
inconsistency of the linear system; (ii) the distance of the initial residual
to the nullspace of the coefficient matrix; (iii) the extremal principal angles
between the ranges of the coefficient matrix and its transpose. These factors
lead to poor conditioning of the extended Hessenberg matrix in the Arnoldi
decomposition and affect the accuracy of the computed least squares solution.
We also compare GMRES with the range restricted GMRES (RR-GMRES) method.
Numerical experiments show typical behaviors of GMRES for small problems with
EP and GP matrices.
