Gradient-based iteration for a class of matrix equations

In this paper, an iterative algorithm is established for solving a class of matrix equations with complex unknowns. By using the hierarchical identification principle, the gradient-based iterative algorithms are constructed to solve the equation AXB + CX^HD = F and the coupled equations A₁XB₁ + A₂X^HB₂ = F₁ and C₁XD₁ + C₂X^HD₂ = F₂. The the convergence factor is presented to guarantee that the iterative algorithms are effective for any initial values. The analysis indicates that if the matrix equation has a unique solution, then the iterative solutions converge to the exact one for any initial value under proper conditions. A numerical example is provided to illustrate the effectiveness of the proposed algorithm.