Inner-iteration Preconditioning with Symmetric Splitting Matrices for Symmetric Singular Linear Systems

Abstract

Abstract. We give conditions such that the conjugate gradient (CG) method and the minimal residual (MINRES) method preconditioned by the stationary inner iterations with symmetric splitting matrices respectively determine a solution of a symmetric linear system including the singular case. These results are applied to the inner-iteration preconditioned CG and MINRES-type methods for solving least squares and minimum-norm solution problems in the rank-deficient case to complement the convergence theory of these methods in [Morikuni, Hayami, SIAM J. Matrix Appl. Anal., 34 (2013), 1–22].