  Matrix equations are omnipresent in (numerical) linear algebra and systems
theory. Especially in model order reduction (MOR) they play a key role in many
balancing based reduction methods for linear dynamical systems. When these
systems arise from spatial discretizations of evolutionary partial differential
equations, their coefficient matrices are typically large and sparse. Moreover,
the numbers of inputs and outputs of these systems are typically far smaller
than the number of spatial degrees of freedom. Then, in many situations the
solutions of the corresponding large-scale matrix equations are observed to
have low (numerical) rank. This feature is exploited by M-M.E.S.S. to find
successively larger low-rank factorizations approximating the solutions. This
contribution describes the basic philosophy behind the implementation and the
features of the package, as well as its application in the model order
reduction of large-scale linear time-invariant (LTI) systems and parametric LTI
systems.
