Numerical solutions to large-scale differential Lyapunov matrix equations
For researchers in numerical linear algebra and control theory, this work provides more efficient computational approaches to a class of matrix equations that arise in large-scale applications.
The paper proposes two new numerical methods for solving large-scale differential Lyapunov matrix equations with low-rank constant terms, demonstrating their effectiveness through theoretical results and numerical experiments.
In the present paper, we consider large-scale differential Lyapunov matrix equations having a low rank constant term. We present two new approaches for the numerical resolution of such differential matrix equations. The first approach is based on the integral expression of the exact solution and an approximation method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix problem is then solved by the Backward Differentiation Formula method (BDF) and the obtained solution is used to build the low rank approximate solution of the original problem. The process being repeated until some prescribed accuracy is achieved. We give some new theoretical results and present some numerical experiments.