Numerical method for solving large-scale differential differential T-Sylvester and nonsymmetric T-Riccati matrix equations

In the paper, we consider differential nonsymmetric T-Riccati equations. So far, it presents an unexplored problem in numerical analysis, theoretical results, and computational methods, which are lacking in the literature. The method we consider is based on an extended block Krylov subspace and Rosenbrock method. The implementation of the algorithm requires only the solution of small-scale differential matrix equations in each time step. Some theoretical results are presented such as an accurate expression of the remaining criteria. The main advantage of the proposed method is that by using this method differential nonsymmetric T-Riccati equation reduces to two T-Sylvester matrix equations. Then report some numerical experiments to show the effectiveness of the proposed method for large-scale problems.