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，对于奇异非一致值域对称线性系统，广义最小残差（GMRES）方法能够在不中断的情况下确定一个最小二乘解。该最小二乘解可能是伪逆解，也可能不是。我们证明采用提升策略可以得到伪逆解。此外，我们针对值域对称线性系统 $\mathbf A\mathbf x=\mathbf b$ 提出一种名为RSMAR（最小化$\mathbf A$-残差）的新型迭代方法。在步骤 $k$ 时，RSMAR在由 $\{\mathbf A, \mathbf r_0\}$ 生成的第 $k$ 个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 $k$ 个残差向量，$\|\cdot\|$ 表示欧几里得向量范数。我们证明，对于值域对称线性系统，RSMAR与GMRES将终止于相同的最小二乘解。我们给出RSMAR的两种实现方案。数值实验表明，对于奇异非一致值域对称线性系统，RSMAR是GMRES类方法中最适用的方法。