New Gradient-Based Iterative-Like Algorithms for Solving a Class of Sylvester Tensor Equations

doi:10.1007/s42967-025-00485-3

Abstract

Abstract: In this paper, we consider deriving some new gradient-based iterative (GI)-like algorithms for solving a class of Sylvester tensor equations, which often arise from control systems and image processing. We first study the optimal parameter and the iteration matrix’s minimal spectral radius of the relaxed GI (RGI) algorithm proposed by Zhang and Wang (Taiwan J Math 26: 501-519, 2022) in terms of matricization of a tensor and straightening operator. Then based on the Jacobi method, by using the diagonal matrices to replace the system matrices in mode products contained in the RGI and the modified RGI (MRGI) algorithms (Taiwan J Math 26: 501-519, 2022), we design the Jacobi RGI (JRGI) and improved MRGI (IMRGI) algorithms for the Sylvester tensor equations, which require less computational load and are more efficient than the RGI and the MRGI ones, respectively. We deduce the sufficient convergence condition and quasi-optimal parameter of the JRGI algorithm, and sufficient and necessary conditions for the convergence of the IMRGI algorithm. Furthermore, we apply a new update strategy to the RGI algorithm and develop an updated RGI (URGI) algorithm for the Sylvester tensor equations, which is different from the MGI (Math Probl Eng 819479: 1-7, 2013) and the MRGI ones. The URGI algorithm takes full advantage of the latest computed results and returns better convergence behavior than RGI, MGI, and MRGI ones. Also, we prove that the proposed URGI algorithm is convergent under proper conditions. Finally, numerical experiments are performed to verify that the proposed algorithms are efficient and have advantages over some existing ones.

Key words: Sylvester tensor equation, Improved modified relaxed gradient-based iterative (IMRGI) algorithm, Jacobi relaxed GI (JRGI) algorithm, Updated relaxed GI (URGI) algorithm, Convergence analysis, Optimal parameter

CLC Number:

65F10

Zhengge Huang. New Gradient-Based Iterative-Like Algorithms for Solving a Class of Sylvester Tensor Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1005-1049.

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