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Volume 35, Issue 5
The Alternating Direction Methods for Solving the Sylvester-Type Matrix Equation $AXB+CX^⊤D=E$

Yifen Ke & Changfeng Ma

DOI: 10.4208/jcm.1608-m2015-0430

J. Comp. Math., 35 (2017), pp. 620-641.

Published online: 2017-10

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  • Abstract

In this paper, we present two alternating direction methods for the solution and best approximate solution of the Sylvester-type matrix equation $AXB+CX^⊤D=E$ arising in the control theory, where $A,B,C,D$ and $E$ are given matrices of suitable sizes. If the matrix equation is consistent (inconsistent), then the solution (the least squares solution) can be obtained. Preliminary convergence properties of the proposed algorithms are presented. Numerical experiments show that the proposed algorithms tend to deliver higher quality solutions with less iteration steps and CPU time than some existing algorithms on the tested problems.

  • Keywords

Sylvester-type matrix equation, Alternating direction method, The least squares solution, Best approximate solution.

  • AMS Subject Headings

65F10, 15A24.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

keyifen2017@163.com (Yifen Ke)

macf@fjnu.edu.cn (Changfeng Ma)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-620, author = {Ke , Yifen and Ma , Changfeng}, title = {The Alternating Direction Methods for Solving the Sylvester-Type Matrix Equation $AXB+CX^⊤D=E$ }, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {5}, pages = {620--641}, abstract = {

In this paper, we present two alternating direction methods for the solution and best approximate solution of the Sylvester-type matrix equation $AXB+CX^⊤D=E$ arising in the control theory, where $A,B,C,D$ and $E$ are given matrices of suitable sizes. If the matrix equation is consistent (inconsistent), then the solution (the least squares solution) can be obtained. Preliminary convergence properties of the proposed algorithms are presented. Numerical experiments show that the proposed algorithms tend to deliver higher quality solutions with less iteration steps and CPU time than some existing algorithms on the tested problems.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1608-m2015-0430}, url = {http://global-sci.org/intro/article_detail/jcm/10035.html} }
TY - JOUR T1 - The Alternating Direction Methods for Solving the Sylvester-Type Matrix Equation $AXB+CX^⊤D=E$ AU - Ke , Yifen AU - Ma , Changfeng JO - Journal of Computational Mathematics VL - 5 SP - 620 EP - 641 PY - 2017 DA - 2017/10 SN - 35 DO - http://doi.org/10.4208/jcm.1608-m2015-0430 UR - https://global-sci.org/intro/article_detail/jcm/10035.html KW - Sylvester-type matrix equation, Alternating direction method, The least squares solution, Best approximate solution. AB -

In this paper, we present two alternating direction methods for the solution and best approximate solution of the Sylvester-type matrix equation $AXB+CX^⊤D=E$ arising in the control theory, where $A,B,C,D$ and $E$ are given matrices of suitable sizes. If the matrix equation is consistent (inconsistent), then the solution (the least squares solution) can be obtained. Preliminary convergence properties of the proposed algorithms are presented. Numerical experiments show that the proposed algorithms tend to deliver higher quality solutions with less iteration steps and CPU time than some existing algorithms on the tested problems.

Ke , Yifen and Ma , Changfeng. (2017). The Alternating Direction Methods for Solving the Sylvester-Type Matrix Equation $AXB+CX^⊤D=E$ . Journal of Computational Mathematics. 35 (5). 620-641. doi:10.4208/jcm.1608-m2015-0430
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