Volume 19, Issue 2
Sine Transform Matrix for Solving Toeplitz Matrix Problems

J. Comp. Math., 19 (2001), pp. 167-176.

Published online: 2001-04

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• Abstract

In recent papers, some authors studied the solutions of symmetric positive definite (SPD) Toeplitz systems $T_nx=b$ by the conjugate gradient method (CG) with different sine transforms based preconditioners. In this paper, we first discuss the properties of eigenvalues for the main known circulant, skew circulant and sine transform based preconditioners. A counter example shows that E. Boman's preconditioner is only positive semi-definite for the banded Toeplitz matrix. To use preconditioner effectively, then we propose a modified Boman's preconditioner and a new Cesaro sum type sine transform based preconditioner. Finally, the results of numerical experimentation with these two preconditioners are presented.

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@Article{JCM-19-167, author = {Cheng , Li-Zhi}, title = {Sine Transform Matrix for Solving Toeplitz Matrix Problems}, journal = {Journal of Computational Mathematics}, year = {2001}, volume = {19}, number = {2}, pages = {167--176}, abstract = {

In recent papers, some authors studied the solutions of symmetric positive definite (SPD) Toeplitz systems $T_nx=b$ by the conjugate gradient method (CG) with different sine transforms based preconditioners. In this paper, we first discuss the properties of eigenvalues for the main known circulant, skew circulant and sine transform based preconditioners. A counter example shows that E. Boman's preconditioner is only positive semi-definite for the banded Toeplitz matrix. To use preconditioner effectively, then we propose a modified Boman's preconditioner and a new Cesaro sum type sine transform based preconditioner. Finally, the results of numerical experimentation with these two preconditioners are presented.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8969.html} }
TY - JOUR T1 - Sine Transform Matrix for Solving Toeplitz Matrix Problems AU - Cheng , Li-Zhi JO - Journal of Computational Mathematics VL - 2 SP - 167 EP - 176 PY - 2001 DA - 2001/04 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8969.html KW - Preconditioner, Toeplitz systems, The fast sine transform, Conjugate gradient algorithm. AB -

In recent papers, some authors studied the solutions of symmetric positive definite (SPD) Toeplitz systems $T_nx=b$ by the conjugate gradient method (CG) with different sine transforms based preconditioners. In this paper, we first discuss the properties of eigenvalues for the main known circulant, skew circulant and sine transform based preconditioners. A counter example shows that E. Boman's preconditioner is only positive semi-definite for the banded Toeplitz matrix. To use preconditioner effectively, then we propose a modified Boman's preconditioner and a new Cesaro sum type sine transform based preconditioner. Finally, the results of numerical experimentation with these two preconditioners are presented.

Li-Zhi Cheng. (1970). Sine Transform Matrix for Solving Toeplitz Matrix Problems. Journal of Computational Mathematics. 19 (2). 167-176. doi:
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