An extended block Golub–Kahan algorithm for large algebraic and differential matrix Riccati equations

Abstract

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.