Brown and Walker (1997) showed that GMRES determines a least squares solution of $ A x = b $ where $ A \in {\bf R}^{n \times n} $ without breakdown for arbitrary $ b, x_0 \in {\bf R}^n $ if and only if $A$ is range-symmetric, i.e. $ {\cal R} (A^{\rm T}) = {\cal R} (A) $, where $ A $ may be singular and $ b $ may not be in the range space ${\cal R} A)$ of $A$. In this paper, we propose applying GMRES to $ A C A^{\rm T} z = b $, where $ C \in {\bf R}^{n \times n} $ is symmetric positive definite. This determines a least squares solution $ x = CA^{\rm T} z $ of $ A x = b $ without breakdown for arbitrary (singular) matrix $A \in {\bf R}^{n \times n}$ and $ b \in {\bf R}^n $. To make the method numerically stable, we propose using the pseudoinverse with an appropriate threshold parameter to suppress the influence of tiny singular values when solving the severely ill-conditioned Hessenberg systems which arise in the Arnoldi process of GMRES when solving inconsistent range-symmetric systems. Numerical experiments show that the method taking $C$ to be the identity matrix gives the least squares solution even when $A$ is not range-symmetric, including the case when $ {\rm index}(A) >1$.

翻译：Brown与Walker（1997）指出，对于任意$b, x_0 \in {\bf R}^n$，GMRES方法在无中断条件下求解$A x = b$的最小二乘解当且仅当$A$为值域对称矩阵，即${\cal R} (A^{\rm T}) = {\cal R} (A)$，其中$A$可能奇异且$b$可能不属于$A$的值空间${\cal R}(A)$。本文提出将GMRES方法应用于$A C A^{\rm T} z = b$，其中$C \in {\bf R}^{n \times n}$为对称正定矩阵。该方法对任意（奇异）矩阵$A \in {\bf R}^{n \times n}$及$b \in {\bf R}^n$，可通过求解$ x = CA^{\rm T} z $获得无中断的最小二乘解。为保障数值稳定性，我们建议在求解GMRES方法Arnoldi过程中产生的严重病态Hessenberg系统时，引入带有适当阈值参数的伪逆算子抑制微小奇异值的影响，该方法特别适用于求解不相容的值域对称系统。数值实验表明，即使$A$不满足值域对称性（包括${\rm index}(A) >1$的情形），取$C$为单位矩阵时仍可获得最小二乘解。