Constraint preconditioners for solving singular saddle point problems

We study constraint preconditioners for solving singular saddle point problems. We analyze properties of the preconditioned matrices, in particular their eigenvalue distributions, and prove that for solving singular saddle point problems by preconditioned GMRES methods with constraint preconditioners, GMRES will determine the least squares solutions at breakdown. In addition, we present some numerical examples to demonstrate the convergence behavior of preconditioned GMRES for solving singular saddle point problems.