  With the ever increasing computational power available and the development of
high-performances computing, investigating the properties of realistic very
large-scale nonlinear dynamical systems has been become reachable. It must be
noted however that the memory capabilities of computers increase at a slower
rate than their computational capabilities. Consequently, the traditional
matrix-forming approaches wherein the Jacobian matrix of the system considered
is explicitly assembled become rapidly intractable. Over the past two decades,
so-called matrix-free approaches have emerged as an efficient alternative. The
aim of this chapter is thus to provide an overview of well-grounded matrix-free
methods for fixed points computations and linear stability analyses of very
large-scale nonlinear dynamical systems.
