Consider applying Krylov subspace methods to systems of linear equations Ax=b or least squares problems min||b-Ax||, where A may be singular and/or nonsymmetric. Let R(A) and N(A) be the range and null space of A, respectively. Brown and Walker gave some conditions concerning R(A) and N(A) for the Generalized Minimal Residual (GMRES) method to converge to a least squares solution without breakdown for singular systems. In this paper, we provide a geometrical view of Krylov subspace methods applied to singular systems by decomposing the algorithm into components of R(A) and its orthogonal complement. Taking coordinates along R(A) and its orthogonal complement will provide an interpretation of the conditions given in Brown and Walker, at the same time giving new proofs for the conditions. We will apply the approach to the GMRES and GMRES(k) methods as well as the Generalized Conjugate Residual (GCR(k)) method, deriving conditions for convergence for inconsistent and consistent singular systems, for each method. Finally, we give examples arising in the finite difference discretization of two-point boundary value problems of an ordinary differential equation as an illustration of the convergence conditions.
Hayami et al. (Thu,) studied this question.