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 and the inverse matrix of the diagonal matrix whose diagonal elements are the diagonal of $A A^{\rm T}$ gives a least squares solution even when $A$ is not range-symmetric, including the case when $ {\rm index}(A) >1$.

翻译：Brown与Walker（1997）证明了：对于任意 $ b, x_0 \in {\bf R}^n $，GMRES能够无中断地确定 $ A x = b $ 的最小二乘解（其中 $ A \in {\bf R}^{n \times n} $）当且仅当 $A$ 是值域对称的，即 $ {\cal R} (A^{\rm T}) = {\cal R} (A) $，此处 $ A $ 可以是奇异的且 $ b $ 可以不在 $A$ 的值域空间 ${\cal R} (A)$ 内。本文提出将GMRES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。该方法能够对任意（奇异）矩阵 $A \in {\bf R}^{n \times n}$ 和 $ b \in {\bf R}^n $ 无中断地确定 $ A x = b $ 的最小二乘解 $ x = CA^{\rm T} z $。为使方法数值稳定，我们建议在求解严重病态的Hessenberg系统时采用带适当阈值参数的伪逆来抑制微小奇异值的影响；此类系统出现在求解非一致值域对称系统时GMRES的Arnoldi过程中。数值实验表明：当取 $C$ 为单位矩阵，或取为对角元素为 $A A^{\rm T}$ 对角线元素的对角矩阵的逆矩阵时，该方法即使在 $A$ 非值域对称（包括 $ {\rm index}(A) >1$ 的情形）时仍能给出最小二乘解。