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 ${\min }_{\mathbf{x}\in {\mathbf{R}}^n}{\Vert \mathbf{b}-A\mathbf{x}{\Vert }_2}^2$, where $A\in {\mathbf{R}}^{n\times n}$ may be singular and $\mathbf{b}\in {\mathbf{R}}^n$, by decomposing the algorithm into the range $\mathcal{R}(A)$ and its orthogonal complement $\mathcal{R}(A{)}^{\perp }$ components. However, we found that the proof of the fact that GMRES gives a least squares solution if $\mathcal{R}(A)=\mathcal{R}(A^T)$ was not complete. In this paper, we will give a complete proof.

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Cited by in corpus (1)

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