Accelerated and Improved Modified Gradient–Based Iterative Algorithms for Solving Sylvester Tensor Equations

Sylvester tensor equation has widely applications in many fields, thus it is meaningful to construct effective methods to solve it. In this paper, we design two new gradient-based iterative-like algorithms for solving the Sylvester tensor equations to further improve computational efficiencies of some existing gradient-based iterative-like ones. By replacing the system matrices in mode products in the modified gradient-based iterative algorithm (Chen, Z. and Lu, L.-Z. [2013] “A gradient based iterative solutions for Sylvester tensor equations,” Math. Probl. Eng. 2013, 151–164) by their diagonal parts, we construct the accelerated modified gradient-based iterative algorithm for the Sylvester tensor equations, which requires less computational load and is more efficient than the modified gradient-based one. Besides, we apply a new updated strategy to the modified gradient-based one and develop an improved modified gradient-based iterative algorithm for the Sylvester tensor equations. Compared with the modified gradient-based one, the improved modified gradient-based iterative algorithm can make more full use of computed results and have better numerical performances. We establish the convergence conditions and convergence intervals of the proposed algorithms based on the spectral radius and matrix spectral norm. Finally, some numerical examples are performed to show that the proposed algorithms are efficient, and outperform several existing gradient-based iterative-like ones in terms of the number of iterations and computational time.