GMRES ON (NEARLY) SINGULAR SYSTEMS

We consider the behavior of the GMRES method for solving a linear syst em Ax = b when A is singular or nearly so, i.e., ill conditioned. The (near) singularity of A may or may not affect the performance of GMRES , depending on the nature of the system and the initial approximate so lution. For singular A, we give conditions under which the GMRES itera tes converge safely to a least-squares solution or to the pseudoinvers e solution. These results also apply to any residual minimizing Krylov subspace method that is mathematically equivalent to GMRES. A practic al procedure is outlined for efficiently and reliably detecting singul arity or ill conditioning when it becomes a threat to the performance of GMRES.