Modified and accelerated relaxed gradient-based iterative algorithms for the complex conjugate and transpose matrix equations
In this paper, by applying the updated technique to the relaxed gradient-based iterative algorithm proposed by Wang et al. (J. Appl. Math. Comput. 67 , 317–341 2021 ), we develop the modified relaxed gradient-based iterative algorithm for the complex conjugate and transpose Sylvester matrix equation...
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| Vydané v: | Numerical algorithms Ročník 97; číslo 4; s. 1955 - 2009 |
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| Jazyk: | English |
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New York
Springer US
01.12.2024
Springer Nature B.V |
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| ISSN: | 1017-1398, 1572-9265 |
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| Abstract | In this paper, by applying the updated technique to the relaxed gradient-based iterative algorithm proposed by Wang et al. (J. Appl. Math. Comput.
67
, 317–341
2021
), we develop the modified relaxed gradient-based iterative algorithm for the complex conjugate and transpose Sylvester matrix equations. Compared with the relaxed gradient-based iterative algorithm, the modified relaxed gradient-based iterative algorithm can make full use of the latest information to compute the next result and lead to a faster convergence rate. Furthermore, in order to reduce the computation cost of each iteration of the relaxed gradient-based iterative algorithm, by replacing the coefficient matrices by their diagonal parts, we construct the accelerated relaxed gradient-based iterative algorithm. By utilizing the properties of the real representation of a complex matrix, matrix norms, and techniques of inequalities, we prove that two proposed iterative algorithms are convergent under proper restrictions. And the quasi-optimal convergence factor of the accelerated relaxed gradient-based iterative algorithm is also derived. Also, some numerical examples are given to show the effectiveness and the superiorities of the proposed algorithms. Lastly, the application in time-varying linear system of the proposed algorithms is presented. |
|---|---|
| AbstractList | In this paper, by applying the updated technique to the relaxed gradient-based iterative algorithm proposed by Wang et al. (J. Appl. Math. Comput. 67, 317–341 2021), we develop the modified relaxed gradient-based iterative algorithm for the complex conjugate and transpose Sylvester matrix equations. Compared with the relaxed gradient-based iterative algorithm, the modified relaxed gradient-based iterative algorithm can make full use of the latest information to compute the next result and lead to a faster convergence rate. Furthermore, in order to reduce the computation cost of each iteration of the relaxed gradient-based iterative algorithm, by replacing the coefficient matrices by their diagonal parts, we construct the accelerated relaxed gradient-based iterative algorithm. By utilizing the properties of the real representation of a complex matrix, matrix norms, and techniques of inequalities, we prove that two proposed iterative algorithms are convergent under proper restrictions. And the quasi-optimal convergence factor of the accelerated relaxed gradient-based iterative algorithm is also derived. Also, some numerical examples are given to show the effectiveness and the superiorities of the proposed algorithms. Lastly, the application in time-varying linear system of the proposed algorithms is presented. In this paper, by applying the updated technique to the relaxed gradient-based iterative algorithm proposed by Wang et al. (J. Appl. Math. Comput. 67 , 317–341 2021 ), we develop the modified relaxed gradient-based iterative algorithm for the complex conjugate and transpose Sylvester matrix equations. Compared with the relaxed gradient-based iterative algorithm, the modified relaxed gradient-based iterative algorithm can make full use of the latest information to compute the next result and lead to a faster convergence rate. Furthermore, in order to reduce the computation cost of each iteration of the relaxed gradient-based iterative algorithm, by replacing the coefficient matrices by their diagonal parts, we construct the accelerated relaxed gradient-based iterative algorithm. By utilizing the properties of the real representation of a complex matrix, matrix norms, and techniques of inequalities, we prove that two proposed iterative algorithms are convergent under proper restrictions. And the quasi-optimal convergence factor of the accelerated relaxed gradient-based iterative algorithm is also derived. Also, some numerical examples are given to show the effectiveness and the superiorities of the proposed algorithms. Lastly, the application in time-varying linear system of the proposed algorithms is presented. |
| Author | Huang, Zhengge Cui, Jingjing |
| Author_xml | – sequence: 1 givenname: Zhengge surname: Huang fullname: Huang, Zhengge email: ZhenggeHuang@163.com organization: College of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Minzu University – sequence: 2 givenname: Jingjing surname: Cui fullname: Cui, Jingjing organization: College of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Minzu University |
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| Keywords | Modified relaxed gradient-based iterative algorithm Optimal convergence factor Complex conjugate and transpose matrix equation 65F10 Real representation Accelerated relaxed gradient-based iterative algorithm Convergence |
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67
, 317–341... In this paper, by applying the updated technique to the relaxed gradient-based iterative algorithm proposed by Wang et al. (J. Appl. Math. Comput. 67, 317–341... |
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