| Citation: |
Fengqing Li, Zhengge Huang, Jingjing Cui, Xiuwen Zheng. DIAGONAL AND NEW MODIFIED GRADIENT-BASED ITERATIVE ALGORITHMS FOR SYLVESTER TENSOR EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 579-635. doi: 10.11948/20240566
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DIAGONAL AND NEW MODIFIED GRADIENT-BASED ITERATIVE ALGORITHMS FOR SYLVESTER TENSOR EQUATIONS
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Abstract
The main objective of this work is to construct some new gradient-based iterative (GI)-like algorithms for solving the (coupled) Sylvester tensor equations. In this paper, we first derive the optimal parameter and the corresponding optimal convergence factor of the GI algorithm (Math. Probl. Eng. 819479 (2013) 1-7) in terms of matricization of a tensor and straightening operator. In order to reduce the computation cost of each iteration of the GI algorithm, enlightened by the idea of the Jacobi method, we replace the system matrices in the GI algorithm by their diagonal parts, and design the diagonal GI (DGI) algorithm for the Sylvester tensor equations. And we derive the sufficient convergence condition, quasi-optimal parameter and quasi-optimal convergence factor of the DGI algorithm. Furthermore, we apply a new update strategy to the GI algorithm and develop the new modified GI (NMGI) algorithm for the Sylvester tensor equations. The proposed NMGI algorithm is different from the MGI one (Math. Probl. Eng. 819479 (2013) 1-7), and can make more full use of the latest computed results and has faster convergence rate than the MGI one for many cases. Also, by utilizing the properties of the tensor norm and techniques of inequalities, we prove that the proposed NMGI algorithm is convergent under proper restrictions. Lastly, some numerical examples are given to validate the efficiencies and advantages of the proposed algorithms for the (coupled) Sylvester tensor equations.
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References
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Fengqing Li, Zhengge Huang, Jingjing Cui, Xiuwen Zheng. DIAGONAL AND NEW MODIFIED GRADIENT-BASED ITERATIVE ALGORITHMS FOR SYLVESTER TENSOR EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 579-635. doi: 10.11948/20240566
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Figure 1.
RES(log10) curves of seven algorithms for Example 6.1 with five values of
$ \eta $ -
Figure 2.
RES(log10) curves of seven algorithms for Example 6.2 with five values of
$ \eta $ $ \rho=2 $ -
Figure 3.
RES(log10) curves of seven algorithms for Example 6.2 with five values of
$ \eta $ $ \rho=3 $ -
Figure 4.
RES(log10) curves of seven algorithms for Example 6.2 with five values of
$ \eta $ $ \rho=5 $ -
Figure 5.
RES(log10) curves of seven algorithms for Example 6.3 with five values of
$ \eta $ $ n=3 $ -
Figure 6.
RES(log10) curves of seven algorithms for Example 6.3 with five values of
$ \eta $ $ n=6 $