On the benefits of the LDLTLDLT factorization for large-scale differential matrix equation solvers

We propose efficient algorithms for solving large-scale matrix differential equations. In particular, we deal with the differential Riccati equations (DRE) and state the applicability to the differential Lyapunov equations (DLE). We focus on methods, based on standard versions of ordinary differential equations, in the matrix setting. The application of these methods yields algebraic Lyapunov equations (ALEs) with a certain structure to be solved in every step. The alternating direction implicit (ADI) algorithm and Krylov subspace based methods allow to exploit this special structure. However, a direct application of classic low-rank formulations requires the use of complex arithmetic. Using an LDLTLDLT-type decomposition of both, the right hand side and the solution of the equation, we avoid this problem. Thus, the proposed methods are a more practical alternative for large-scale problems arising in applications. Also, they make the application of higher order methods feasible. The numerical results show the better performance of the proposed methods compared to earlier formulations.