  A modification of the generalized shift-splitting (GSS) method is presented
for solving singular saddle point problems. In this kind of modification, the
diagonal shift matrix is replaced by a block diagonal matrix which is symmetric
positive definite. Semi-convergence of the proposed method is investigated. The
induced preconditioner is applied to the saddle point problem and the
preconditioned system is solved by the restarted generalized minimal residual
method. Eigenvalue distribution of the preconditioned matrix is also discussed.
Finally some numerical experiments are given to show the effectiveness and
robustness of the new preconditioner. Numerical results show that the modified
GSS method is superior to the classical GSS method.
