On Hessenberg type methods for low-rank Lyapunov matrix equations

The Hessenberg process can be seen as an alternative to the well-known Arnoldi and Lanczos processes. Block and global versions of the Hessenberg process were used for solving linear systems with multiple right-hand sides and large Sylvester matrix equations. In this work, we describe two new Hessenberg based methods for obtaining approximate solutions to low-rank Lyapunov matrix equations. The first one is based on a Ruhe variant of the block Hessenberg process and belongs to the class of block Krylov solvers. The second one is a matrix Krylov solver and uses the global Hessenberg process. The numerical comparisons we have made show that solving the Lyapunov equation via the global Hessenberg process gives better results compared to the use of the block Hessenberg process.