The solution of the large-scale differential Sylvester matrix problems by extended nonsymmetric block Lanczos algorithm

In this article, we present two novel methodologies utilizing the extended nonsymmetric block Lanczos method for the efficient resolution of large-scale differential Sylvester equations. The first approach employs the approximation of the matrix exponential to compute the integral representation of the exact solution. The second approach involves projecting the initial problem onto an extended block Krylov subspaces Kme(P,V) and Kme(PT,W), resulting in a reduced-dimensional differential Sylvester matrix equation. The latter equation is then solved using the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), yielding a low-rank approximate solution to the original problem. Theoretical results are provided, along with numerical experiments demonstrating the efficacy of the proposed methods.