GMRES on singular systems revisited

09/01/2020

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by   Ken Hayami, et al.

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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 - A x_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.