In this paper, combining the precondition technique and momentum item with the gradient-based iteration algorithm, two accelerated iteration algorithms are presented for solving the Sylvester matrix equation $AX+XB=C$. Sufficient conditions to guarantee the convergence properties of the proposed algorithms are analyzed in detail. Varying the parameters of these algorithms in each iteration, the corresponding adaptive iteration algorithms are also provided, and the adaptive parameters can be explicitly obtained by the minimum residual technique. Several numerical examples are implemented to illustrate the effectiveness of the proposed algorithms.

By combining the hierarchical identification principle with HSS splitting, we presented the HSS splitting hierarchical identification algorithm for solving the Sylvester matrix equation in this paper. To enhance the convergence rate of the algorithm, the momentum item was introduced in the iteration. We conducted an in-depth analysis of the sufficient conditions that ensured the convergence properties of the proposed algorithms. Additionally, the optimal parameters involved in the algorithms were computed exactly in each iteration by the minimum residual technique for specific cases. Thus, the adaptive forms of the corresponding algorithms were obtained. Finally, several numerical examples were implemented to demonstrate the superiority and effectiveness of the designed algorithms in this paper.