  We present an implementation of an efficient algorithm for the calculation of
the spectrum of one-dimensional quantum systems with periodic boundary
conditions. This algorithm is based on a matrix product representation for
quantum states (MPS), and a similar representation for Hamiltonians and other
operators (MPO). It is significantly more efficient for systems of about 100
sites and more than for small quantum systems. We apply the formalism to
calculate the ground state and first excited state of a spin-1 Heisenberg ring
and deduce the size of the Haldane gap. The results are compared to previous
high-precision DMRG calculations. Furthermore, we study spin-1 systems with a
biquadratic nearest-neighbor interaction and show first results of an
application to a mesoscopic Hubbard ring of spinless Fermions which carries a
persistent current.
