THE EFFECTIVE AND MODIFIED JACOBI GRADIENT BASED ITERATIVE ALGORITHMS FOR THE DISCRETE-TIME PERIODIC SYLVESTER MATRIX EQUATIONS

Xiaowen Wu; Zhengge Huang

doi:10.11948/20240393

2025 Volume 15 Issue 4

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Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(4): 2044-2088

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Xiaowen Wu, Zhengge Huang. THE EFFECTIVE AND MODIFIED JACOBI GRADIENT BASED ITERATIVE ALGORITHMS FOR THE DISCRETE-TIME PERIODIC SYLVESTER MATRIX EQUATIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2044-2088. doi: 10.11948/20240393

Citation:

Xiaowen Wu, Zhengge Huang. THE EFFECTIVE AND MODIFIED JACOBI GRADIENT BASED ITERATIVE ALGORITHMS FOR THE DISCRETE-TIME PERIODIC SYLVESTER MATRIX EQUATIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2044-2088. doi: 10.11948/20240393

THE EFFECTIVE AND MODIFIED JACOBI GRADIENT BASED ITERATIVE ALGORITHMS FOR THE DISCRETE-TIME PERIODIC SYLVESTER MATRIX EQUATIONS

Xiaowen Wu^1,,
Zhengge Huang^1, ,

1.
School of Mathematical Sciences, Center for Applied Mathematics of Guangxi, Guangxi Minzu University, 530006 Nanning, China

Author Bio: Email: xiaowenwu1998@163.com(X. Wu)
Corresponding author: Email: ZhenggeHuang@163.com(Z. Huang)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 12361078) and the Guangxi Natural Science Foundation (No. 2024GXNSFAA010498)

Abstract

Abstract

In this paper, we discuss the new convergence properties of some gradient based iterative (GI) algorithms and propose two new GI-like algorithms for solving the discrete-time periodic Sylvester (DTPS) matrix equations and its generalized version, which often arise in the fields of physics, medicine and so forth. We first review the Jacobi GI (JGI) and accelerated JGI (AJGI) algorithms (Appl. Numer. Math., 168 (2021) 251-273) for the DTPS matrix equations, and establish the new and correct convergence conditions of these two algorithms. Then we apply a new update strategy to the JGI algorithm and develop the effective Jacobi gradient based iterative (EJGI) algorithm for solving the DTPS matrix equations, which is different from the AJGI one. Furthermore, based on the ideas of the JGI and the Gauss-Seidel (G-S) algorithms, we construct the modified Jacobi gradient based iterative (MJGI) algorithm for the generalized discrete-time periodic Sylvester (GDTPS) matrix equations. Compared with the JGI algorithm, the MJGI algorithm can make full use of the latest information to compute the next result and lead to a faster convergence rate. By utilizing the properties of the matrix norms, Kronecker product and techniques of inequalities, we prove that two proposed iterative algorithms are convergent under proper restrictions. Finally, some numerical examples are given to validate the efficiencies and advantages of the proposed EJGI and MJGI algorithms for DTPS and GDTPS matrix equations.
- DTPS matrix equations /
- GDTPS matrix equations /
- EJGI algorithm /
- MJGI algorithm /
- convergence properties
MSC: 65F10, 65H10

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References

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