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The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices

  Published:2017-01-12
 Printed: Dec 2017
  • Zhongyun Liu,
    School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410076, P.R. China
  • Xiaorong Qin,
    Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal
  • Nianci Wu,
    Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal
  • Yulin Zhang,
    Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal
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Abstract

It is known that every Toeplitz matrix $T$ enjoys a circulant and skew circulant splitting (denoted by CSCS) i.e., $T=C-S$ with $C$ a circulant matrix and $S$ a skew circulant matrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss-Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant $\alpha$ such that the shifted CSCS iteration converges much faster than the Gauss-Seidel iteration, no matter whether the CSCS itself is convergent or not.
Keywords: Hermitian positive definite, CSCS splitting, Gauss-Seidel splitting, iterative method, Toeplitz matrix Hermitian positive definite, CSCS splitting, Gauss-Seidel splitting, iterative method, Toeplitz matrix
MSC Classifications: 15A23, 65F10, 65F15 show english descriptions Factorization of matrices
Iterative methods for linear systems [See also 65N22]
Eigenvalues, eigenvectors
15A23 - Factorization of matrices
65F10 - Iterative methods for linear systems [See also 65N22]
65F15 - Eigenvalues, eigenvectors
 

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