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An LDLT factorization based ADI algorithm for solving large‐scale differential matrix equations

Large‐scale differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time varying systems etc. Here, we will focus on matrix Riccati differential equations (RDE). Solving such matrix valued ordinary differential equations (ODE) is a highly storage and time consuming process. Therefore, it is necessary to develop efficient solution strategies minimizing both. We present an LDLT factorization based ADI method for solving algebraic Lyapunov equations (ALE) arising in the innermost iteration during the application of Rosenbrock ODE solvers to RDEs. We show that the LDLT‐type decomposition avoids complex arithmetic, as well as cancellation effects arising from indefinite right hand sides of the ALEs appearing in the classic ZZT based approach. Additionally, a certain number of linear system solves can be saved within the ADI algorithm by reducing the number of column blocks in the right hand sides while the full accuracy of the standard low‐rank ADI is preserved. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)