On $${\varvec{m}}$$ m -step Hermitian and skew-Hermitian splitting preconditioning methods

Abstract

Preconditioners are often applied in various Krylov subspace iteration methods to improve their computing efficiency. In this paper, we consider solving linear systems with a non-Hermitian positive definite coefficient matrix using preconditioned Krylov subspace iteration methods such as the generalized minimal residual (GMRES) method. An $$m$$ -step polynomial preconditioner is designed based on the Hermitian and skew-Hermitian splitting (HSS) iteration method proposed by Bai et al. (SIAM J Matrix Anal Appl 24:603–626, 2003). The proposed preconditioned system is solved by fully utilizing the HSS iteration method. Theoretical and experimental results show that the proposed $$m$$ -step preconditioner is efficient in accelerating GMRES for solving a non-Hermitian positive definite linear system.