  The Deligne--Simpson problem is an existence problem for connections with
specified local behavior. Almost all previous work on this problem has
restricted attention to connections with regular or unramified singularities.
Recently, the authors, together with Kulkarni and Matherne, formulated a
version of the Deligne--Simpson problem where certain ramified singular points
are allowed and solved it for the case of Coxeter connections, i.e.,
connections on the Riemann sphere with a maximally ramified singularity at zero
and (possibly) an additional regular singular point at infinity. A certain
matrix completion problem, which we call the Upper Nilpotent Completion
Problem, plays a key role in our solution. This problem was solved by Krupnik
and Leibman, but their work does not provide a practical way of constructing
explicit matrix completions. Accordingly, our previous work does not give
explicit Coxeter connections with specified singularities. In this paper, we
provide a numerically stable and highly efficient algorithm for producing upper
nilpotent completions of certain matrices that arise in the theory of Coxeter
connections. Moreover, we show how the matrices generated by this algorithm can
be used to provide explicit constructions of Coxeter connections with arbitrary
unipotent monodromy in each case that such a connection exists.
