NR-SSOR right preconditioned RRGMRES for arbitrary singular systems and least squares problems

GMRES is known to determine a least squares solution of $A x = b$ where $A \in R^{n \times n}$ without breakdown for arbitrary $b \in R^n$, and initial iterate $x_0 \in R^n$ if and only if $A$ is range-symmetric, i.e. $R(A^T) = R(A)$, where $A$ may be singular and $b$ may not be in the range space $R(A)$ of $A$. In this paper, we propose applying the Range Restricted GMRES (RRGMRES) to $A C A^T z = b$, where $C \in R^{n \times n}$ is symmetric positive definite. This determines a least squares solution $x = C A^T z$ of $A x = b$ without breakdown for arbitrary (singular) matrix $A \in R^{n \times n}$ and $b, x_0 \in R^n$, and is much more stable and accurate compared to GMRES, RRGMRES and MINRES-QLP applied to $A x = b$ for inconsistent problems when $b \notin R(A)$. In particular, we propose applying the NR-SSOR as the inner iteration right preconditioner, which also works efficiently for least squares problems $\min_{x \in R^n} \| b - A x\|_2$ for $A \in R^{m \times n}$ and arbitrary $b \in R^m$. Numerical experiments demonstrate the validity of the proposed method.