Nested splitting conjugate gradient method for matrix equation AXB=CAXB=C and preconditioning

In this paper, we present a nested splitting conjugate gradient (NSCG) iteration method for solving a class of matrix equations with nonsymmetric coefficient matrices. This method is actually inner/outer iterations, which employs a CG-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrices. Convergence conditions of this method are studied in depth and numerical experiments show the efficiency of this method. Moreover, we show that the use of the quasi-Hermitian splitting as a preconditioner can induce an accurate, robust and effective preconditioned Krylov subspace method.