On the alternating SHSS iteration method for solving the matrix equation AXB = C

Abstract

The Hermitian and skew-Hermitian splitting (HSS) method is a simple yet powerful technique for solving the linear system Ax = b and some matrix equations, such as matrix equation AXB = C and Sylvester matrix equation AX + XB = C. In this work, we propose an alternating single-step HSS (ASHSS) framework for solving the matrix equation AXB = C. By inserting a single-step preconditioned HSS step and exploiting an alternating update, only two Hermitian positive-definite matrix equations are needed to be solved at each iteration, and the analysis of convergence property becomes notably simpler. The new formulation also admits a broad class of parameter matrices, yielding three concrete variants whose optimal parameters are derived analytically. Numerical experiments demonstrate that the proposed algorithms outperform the GI and HSS method in both iteration count and CPU time.