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

Abstract

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 $\mathbf{Ax}=\mathbf{b}$. At step $k$ RSMAR minimizes $\Vert \mathbf{A}{\mathbf{r}}_k\Vert$ in the $k$th Krylov subspace generated with $\{\mathbf{A},{\mathbf{r}}_0\}$ rather than $\Vert {\mathbf{r}}_k\Vert$, where ${\mathbf{r}}_k$ is the $k$th residual vector and $\Vert \cdot \Vert$ 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.

22 pages, 4 figures