Circulant and skew-circulant splitting methods for Toeplitz systems
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 6th Japan--China joint seminar on numerical mathematics, university of Tsukuba, Japan, 5-9 August 2002
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On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation AXB=C
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Journal of Computational and Applied Mathematics
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We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of a three-dimensional convection-diffusion equation is used to illustrate the advantages of our methods.