  The Coxeter lattices, which we denote $A_{n/m}$, are a family of lattices
containing many of the important lattices in low dimensions. This includes
$A_n$, $E_7$, $E_8$ and their duals $A_n^*$, $E_7^*$ and $E_8^*$. We consider
the problem of finding a nearest point in a Coxeter lattice. We describe two
new algorithms, one with worst case arithmetic complexity $O(n\log{n})$ and the
other with worst case complexity O(n) where $n$ is the dimension of the
lattice. We show that for the particular lattices $A_n$ and $A_n^*$ the
algorithms reduce to simple nearest point algorithms that already exist in the
literature.
