  We study the properties of excited states in one-dimensional many-body
localized (MBL) systems using a matrix product state algorithm. First, the
method is tested for a large disordered non-interacting system, where for
comparison we compute a quasi-exact reference solution via a Monte Carlo
sampling of the single-particle levels. Thereafter, we present extensive data
obtained for large interacting systems of L~100 sites and large bond dimensions
chi~1700, which allows us to quantitatively analyze the scaling behavior of the
entanglement S in the system. The MBL phase is characterized by a logarithmic
growth (L)~log(L) over a large scale separating the regimes where volume and
area laws hold. We check the validity of the eigenstate thermalization
hypothesis. Our results are consistent with the existence of a mobility edge.
