Symmetric inner-iteration preconditioning for rank-deficient least squares problems

Stationary iterative methods with a symmetric splitting matrix are performed as inner-iteration preconditioning for Krylov subspace methods. We give a necessary and sufficient condition such that the inner-iteration preconditioning matrix is definite, and show that conjugate gradient (CG) method preconditioned by the inner iterations determines a solution of a symmetric and positive semidefinite linear system, and the minimal residual (MR) method preconditioned by the inner iterations determines a solution of a symmetric linear system including the singular case. These results are applied to the CG and MR-type methods such as the CGLS, LSMR, and CGNE methods preconditioned by inner iterations, and thus justify using these methods for solving least squares and minimum-norm solution problems those coefficient matrices are not necessarily of full rank. Thus, we complement the theory of these methods presented in [K. Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 34 (2013), pp. 1-22], [K. Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 36 (2015), pp. 225-250], and give bounds for these methods.