Subspace Correction Methods for Singular Systems

We investigate the application of minimal residual and orthogonal residualsubspace correction methods to singular linear systems Ax = b. Special em-phasis is put on the special case of Krylov subspace methods. If A has index 1(i.e., if all Jordan blocks associated with the eigenvalue λ = 0 of A are 1× 1)the behaviour of these iterative methods is well understood (see, e.g., [1] and[2]). We here describe, for the case of an arbitrary index, the break-downsof these methods as well as the ”limit” of the associated iterates. We furthercharacterize the situations where this limit represents a (least squares) solutionof Ax = b. References[1] Peter N. Brown and Homer F. Walker. GMRES on (nearly) singularsystems. SIAM J. Matrix Anal. Appl. 18, 37–51 (1997).[2] Ken Hayami. On the behaviour of the conjugate residual method forsingular systems. Proceedings of the Fifth China-Japan Joint Seminar onNumerical Analysis, Science Press, Beijing 2002.