In this paper we propose a new projection method to solve both large-scale continuous-time matrix Riccati equations and differential matrix Riccati equations. The new approach projects the problem onto an extended block Krylov subspace and gets a low-dimensional equation. We use the block Golub–Kahan procedure to construct the orthonormal bases for the extended Krylov subspaces. For matrix Riccati equations, the reduced problem is then solved by means of a direct Riccati scheme such as the Schur method. When we solve differential matrix Riccati equations, the reduced problem is solved by the Backward Differentiation Formula (BDF) method and the obtained solution is used to build the low rank approximate solution of the original problem. Finally, we give some theoretical results and present numerical experiments.

在本文中，我们提出了一种新的投影方法来求解大型连续时间矩阵Riccati方程和微分矩阵Riccati方程。新方法将问题投影到扩展的块Krylov子空间上，并得到一个低维方程。我们使用块Golub–Kahan程序来构造扩展Krylov子空间的正交基。然后，对于矩阵Riccati方程，可通过直接Riccati方案（例如Schur方法）解决简化问题。当我们求解微分矩阵Riccati方程时，可通过向后微分公式（BDF）方法求解简化问题，并将获得的解用于构建原始问题的低秩近似解。最后，我们给出一些理论结果并进行数值实验。