  We consider the low-rank alternating directions implicit (ADI) iteration for
approximately solving large-scale algebraic Sylvester equations. Inside every
iteration step of this iterative process a pair of linear systems of equations
has to be solved. We investigate the situation when those inner linear systems
are solved inexactly by an iterative methods such as, for example,
preconditioned Krylov subspace methods. The main contribution of this work are
thresholds for the required accuracies regarding the inner linear systems which
dictate when the employed inner Krylov subspace methods can be safely
terminated. The goal is to save computational effort by solving the inner
linear system as inaccurate as possible without endangering the functionality
of the low-rank Sylvester-ADI method. Ideally, the inexact ADI method mimics
the convergence behaviour of the more expensive exact ADI method, where the
linear systems are solved directly. Alongside the theoretical results, also
strategies for an actual practical implementation of the stopping criteria are
developed. Numerical experiments confirm the effectiveness of the proposed
strategies.
