Most numerical integration algorithms are not designed specifically for
Hamiltonian systems and do not respect their characteristic properties, which
include the preservation of phase space volume with time. This can lead to
spurious damping or excitation. Methods that do preserve all the Hamiltonian
properties, i.e., for which the time-forward map is symplectic, are called
symplectic integration algorithms or SIAs. Although such integrators are
symplectic in theory, they are not symplectic if implemented using
finite-precision arithmetic. This paper explains how to eliminate this problem
by using ``Lattice SIAs'' and shows that these methods yield significant
advantages when the computational error is dominated by roundoff. Using a
lattice SIA is equivalent to evolving the exact solution of a problem with a
Hamiltonian that is slightly different from the original. Lattice methods are
useful for studies of the long-term evolution of Hamiltonian dynamical systems.