It is well known that for singular inconsistent range-symmetric linear systems, the generalized minimal residual (GMRES) method determines a least squares solution without breakdown. The reached least squares solution may be or not be the pseudoinverse solution. We show that a lift strategy can be used to obtain the pseudoinverse solution. In addition, we propose a new iterative method named RSMAR (minimum $\mathbf A$-residual) for range-symmetric linear systems $\mathbf A\mathbf x=\mathbf b$. At step $k$ RSMAR minimizes $\|\mathbf A\mathbf r_k\|$ in the $k$th Krylov subspace generated with $\{\mathbf A, \mathbf r_0\}$ rather than $\|\mathbf r_k\|$, where $\mathbf r_k$ is the $k$th residual vector and $\|\cdot\|$ denotes the Euclidean vector norm. We show that RSMAR and GMRES terminate with the same least squares solution when applied to range-symmetric linear systems. We provide two implementations for RSMAR. Our numerical experiments show that RSMAR is the most suitable method among GMRES-type methods for singular inconsistent range-symmetric linear systems.

翻译：众所周知，对于奇异不相容值域对称线性系统，广义最小残差法（GMRES）能够无中断地确定最小二乘解。所得到的最小二乘解可能是也可能不是伪逆解。我们证明可以采用提升策略来获得伪逆解。此外，我们针对值域对称线性系统$\mathbf A\mathbf x=\mathbf b$提出一种新的迭代方法RSMAR（最小$\mathbf A$-残差法）。在步骤$k$中，RSMAR在由$\{\mathbf A, \mathbf r_0\}$生成的$k$阶Krylov子空间中最小化$\|\mathbf A\mathbf r_k\|$而非$\|\mathbf r_k\|$，其中$\mathbf r_k$为第$k$步残差向量，$\|\cdot\|$表示欧几里得向量范数。我们证明当应用于值域对称线性系统时，RSMAR与GMRES以相同的最小二乘解终止。我们提供了RSMAR的两种实现方式。数值实验表明，对于奇异不相容值域对称线性系统，RSMAR是GMRES类方法中最适用的方法。