  Projected Entangled Pair States (PEPS) are a promising ansatz for the study
of strongly correlated quantum many-body systems in two dimensions. But due to
their high computational cost, developing and improving PEPS algorithms is
necessary to make the ansatz widely usable in practice. Here we analyze several
algorithmic aspects of the method. On the one hand, we quantify the connection
between the correlation length of the PEPS and the accuracy of its approximate
contraction, and discuss how purifications can be used in the latter. On the
other, we present algorithmic improvements for the update of the tensor that
introduce drastic gains in the numerical conditioning and the efficiency of the
algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with
the Heisenberg and quantum Ising models on lattices of up to $21 \times 21$
sites.
