GMRES on singular systems revisited

Abstract

In [Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18:449-469], the authors analyzed the convergence behaviour of the Generalized Minimal Residual (GMRES) method for the least squares problem min x∈R n b − Ax 2 2 , where A ∈ R n×n may be singular and b ∈ R n , by decomposing the algorithm into the range R(A) and its orthogonal complement R(A) ⊥ components. However, we found that the proof of the fact that GMRES gives a least squares solution if R(A) = R(A T ) was not complete. In this paper, we will give a complete proof.