  The density matrix renormalization group (DMRG) method generates the
low-energy states of linear systems of $N$ sites with a few degrees of freedom
at each site by starting with a small system and adding sites step by step
while keeping constant the dimension of the truncated Hilbert space. DMRG
algorithms are adapted to open chains with inversion symmetry at the central
site, to cyclic chains and to weakly coupled chains. Physical properties rather
than energy accuracy is the motivation. The algorithms are applied to the edge
states of linear Heisenberg antiferromagnets with spin $S \ge 1$ and to the
quantum phases of a frustrated spin-1/2 chain with exchange between first and
second neighbors. The algorithms are found to be accurate for extended Hubbard
and related 1D models with charge and spin degrees of freedom.
