Two accelerated gradient-based iteration methods for solving the Sylvester matrix equation AX + XB = C

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.