  We present a new high-order coupled cluster method (CCM) formalism for the
ground states of lattice quantum spin systems for general spin quantum number,
$s$. This new ``general-$s$'' formalism is found to be highly suitable for a
computational implementation, and the technical details of this implementation
are given. To illustrate our new formalism we perform high-order CCM
calculations for the one-dimensional spin-half and spin-one antiferromagnetic
{\it XXZ} models and for the one-dimensional spin-half/spin-one ferrimagnetic
{\it XXZ} model. The results for the ground-state properties of the isotropic
points of these systems are seen to be in excellent quantitative agreement with
exact results for the special case of the spin-half antiferromagnet and results
of density matrix renormalisation group (DMRG) calculations for the other
systems. Extrapolated CCM results for the sublattice magnetisation of the
spin-half antiferromagnet closely follow the exact Bethe Ansatz solution, which
contains an infinite-order phase transition at $\Delta=1$. By contrast,
extrapolated CCM results for the sublattice magnetisation of the spin-one
antiferromagnet using this same scheme are seen to go to zero at $\Delta
\approx 1.2$, which is in excellent agreement with the value for the onset of
the Haldane phase for this model. Results for sublattice magnetisations of the
ferrimagnet for both the spin-half and spin-one spins are non-zero and finite
across a wide range of $\Delta$, up to and including the Heisenberg point at
$\Delta=1$.
