on some numerical methods for solving large-scale differential t-lyapunov matrix equations

In this paper, we present two new approaches to solve large-scale differential t-lyapunov matrix equations. the first one is based on the extended block krylov subspaces, and the second is based on the extended global krylov subspaces. the initial problem is projected onto an extended block (or global) krylov subspaces to get a small-scale differential t-lyapunov matrix equation. the latter problem is solved by iterative methods (rosenbrock or bdf method), then the obtained solution is used to create a low-rank approximate solution of the original problem. this process is being replicated, which increases the dimension of the projection space until some planned accuracy is achieved. we give some new theoretical results and numerical experiments then we compare the new approaches.