On the HSS iteration methods for positive definite Toeplitz linear systems

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, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003) 603–626], 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 = 1 2 ( A + A T ) is a centrosymmetric matrix and the skew-symmetric part S = 1 2 ( A − A 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.