Computational Krylov-based methods for large-scale differential Sylvester matrix problems
For researchers in numerical linear algebra, this work offers incremental improvements in solving a specific class of matrix equations.
This paper proposes two Krylov-based methods for solving large-scale differential Sylvester matrix equations with low-rank constant terms, providing theoretical error bounds and residual norm expressions. Numerical experiments compare the two approaches.
In the present paper, we propose Krylov-based methods for solving large-scale differential Sylvester matrix equations having a low rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov 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 Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as BDF or Rosenbrock method and the obtained solution is used to build the low rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.