  The rational Krylov subspace method (RKSM) and the low-rank alternating
directions implicit (LR-ADI) iteration are established numerical tools for
computing low-rank solution factors of large-scale Lyapunov equations. In order
to generate the basis vectors for the RKSM, or extend the low-rank factors
within the LR-ADI method the repeated solution to a shifted linear system is
necessary. For very large systems this solve is usually implemented using
iterative methods, leading to inexact solves within this inner iteration. We
derive theory for a relaxation strategy within these inexact solves, both for
the RKSM and the LR-ADI method. Practical choices for relaxing the solution
tolerance within the inner linear system are then provided. The theory is
supported by several numerical examples.
