  We discuss the numerical solution methods available when solving for the
steady-state density matrix of a time-independent open quantum optical system,
where the system operators are expressed in a suitable basis representation as
sparse matrices. In particular, we focus on the difficulties posed by the
non-Hermitian structure of the Lindblad super operator, and the numerical
techniques designed to mitigate these pitfalls. In addition, we introduce a
doubly iterative inverse-power method that can give reduced memory and runtime
requirements in situations where other iterative methods are limited due to
poor bandwidth and profile reduction. The relevant methods are demonstrated on
several prototypical quantum optical systems where it is found that iterative
methods based on iLU factorization using reverse Cuthill-Mckee ordering tend to
outperform other solution techniques in terms of both memory consumption and
runtime as the size of the underlying Hilbert space increases. For eigenvalue
solving, Krylov iterations using the stabilized bi-conjugate gradient method
outperform generalized minimal residual methods. In contrast, minimal residual
methods work best for solvers based on direct LU decomposition. This work
serves as a guide for solving the steady-state density matrix of an arbitrary
quantum optical system, and points to several avenues of future research that
will extend the applicability of these classical algorithms in absence of a
quantum computer.
