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On the HSS iteration methods for positive definite Toeplitz linear systems. (English) Zbl 1185.65055
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 $H=\frac{1}{2}\left(A+{A}^{\text{T}}\right)$ is a centrosymmetric matrix and the skew-symmetric part $S=\frac{1}{2}\left(A-{A}^{\text{T}}\right)$ 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.

MSC:
 65F10 Iterative methods for linear systems