| Differential matrix equations are widely used in many fields.Among the more important is the differential Lyapunov equation and differential Riccati equation,the two types of linear and nonlinear matrix equation is one of the important subjects of science and engineering,are applied to the optimal control theory,dynamic programming,model reduction of linear dynamic system,statistics filter,and other fields.Because the analytic solution of matrix differential equation is difficult to obtain,its numerical solution rule has become the focus of many scholars.The modified Douglas splitting method is a new method for solving differential equations based on splitting thoughts.The complex function on the right side of the equation splits into several simple functions,which is easy to calculate quickly and has good stability.In this paper,by applying this method to the differential matrix equation,the Modified douglas splitting scheme for the differential matrix equation is obtained by using the natural three-term splitting of the equations.The implementation of this algorithm only needs to solve a linear algebraic system with multiple right-hand sides in each time step,it has good stability and high accuracy.Then,we study the solution scheme of differential Lyapunov equation and differential Riccati equation,and prove that the method has second order accuracy.Meanwhile,we also prove that the method keep symmetric semi-positive quality of the solution of differential Lyapunov equation.In addition,based on the proposed modified Douglas splitting method,we also constructed low-rank algorithm of the large-scale differential Lyapunov equation and differential Riccati equation,which modified the calculation and storage efficiency.Finally,the low-rank algorithm of the theoretical prior error analysis,numerical results verified the correctness of the theoretical analysis.The main content of this paper is as follows:In the first part,we study the modified Douglas splitting algorithm for differential matrix equations,prove the second order convergence of the full rank method and the symmetric semi-positive properties of the solutions of differential Lyapunov equations.In this part,through the natural splitting of the equation,the modified Douglas splitting scheme is used to iteratively solve the numerical solution of the differential equation.The second order convergence of the method and the preservation structure of the solution scheme of differential Lyapunov equation are proved.The algorithm only needs to solve a linear algebra system with multiple right-side terms at each time step,and does not need to solve the algebraic Lyapunov equation at each time step.Finally,numerical experiments are carried out to compare with the midpoint formula and Rosenbrock method to illustrate the effectiveness of the algorithm.In the second part,the low-rank modified Douglas splitting methods for differential Lyapunov equation and differential Riccati equation is put forward,and the convergence of the low-rank algorithm is proved.The numerical solutions of differential Lyapunov equation and differential Riccati equation are obtained based on the full rank method.By using QR decomposition and singular value decomposition to compress the numerical solutions,the low-rank solution of the algorithm is obtained.It is proved that the convergence order of the low-rank algorithm is 2.Finally,this paper gives numerical experiments to prove the convergence of the method,analogy to the modified Douglas splitting algorithm with full rank,and highlight the superiority of the low-rank modified Douglas splitting algorithm. |
