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arXiv · 2009.00371

GMRES on singular systems revisited

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

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Ken Hayami, Kota Sugihara. 2020-09-01. GMRES on singular systems revisited. https://doi.org/10.1002/nla.2368

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