  We study the cone of completely positive (cp) matrices for the first
interesting case $n = 5$. This is a semialgebraic set, which means that the
polynomial equalities and inequlities that define its boundary can be derived.
We characterize the different loci of this boundary and we examine the two open
sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and
able to compute the cp-factorization even for matrices in the boundary. With
our results, many new example cases can be produced and several insightful
numerical experiments are performed that illustrate the difficulty of the
cp-factorization problem.
