Obtaining the pseudoinverse solution of singular range-symmetric linear systems with GMRES-type methods

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.