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In this paper,a generalized preconditioned Hermitian and skew-Hermitian splitting (GPHSS) iteration method for a non-Hermitian positive-definite matrix is studied,which covers standard Hermitian and skew-Hermitian splitting (HSS) iteration and also many existing variants.Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix.From practical point of view,we have analyzed and implemented inexact generalized preconditioned Hermitian and skew-Hermitian splitting (IGPHSS) iteration,which employs Krylov subspace methods as its inner processes.Numerical experiments from three-dimensional convection-diffusion equation show that the GPHSS and IGPHSS iterations are efficient and competitive with standard HSS iteration and AHSS iteration.