Summary: We study the HSS iteration method for large sparse non-Hermitian positive definite Toeplitz linear systems, which first appears in Bai, Golub and Ng’s paper published in 2003 [

*Z.-Z. Bai, G. H. Golub* and

*M. K. Ng*, SIAM J. Matrix Anal. Appl. 24, No. 3, 603–626 (2003;

Zbl 1036.65032)], and HSS stands for the Hermitian and skew-Hermitian splitting of the coefficient matrix

$A$. In this note we use the HSS iteration method based on a special case of the HSS splitting, where the symmetric part

$H=\frac{1}{2}(A+{A}^{\text{T}})$ is a centrosymmetric matrix and the skew-symmetric part

$S=\frac{1}{2}(A-{A}^{\text{T}})$ is a skew-centrosymmetric matrix for a given Toeplitz matrix. Hence, fast methods are available for computing the two half-steps involved in the HSS and IHSS iteration methods. Some numerical results illustrate their effectiveness.