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.

翻译：Deligne–Simpson问题是一个关于具有指定局部行为的联络的存在性问题。此前几乎所有相关工作均局限于正则奇点或无分歧奇点的联络。近期，作者与Kulkarni及Matherne共同提出了允许某些分歧奇点的Deligne–Simpson问题版本，并针对Coxeter联络（即黎曼球面上在零点具有极大分歧奇点、且在无穷远点可能具有额外正则奇点的联络）情形给出了解答。在我们的解答中，一个被称为“上幂零补全问题”的矩阵补全问题起到了关键作用。该问题虽已由Krupnik和Leibman解决，但他们的工作未提供构造显式矩阵补全的实用方法。因此，我们此前的工作未能给出具有指定奇点的显式Coxeter联络。本文提出了一种数值稳定且高效的上幂零补全算法，可生成Coxeter联络理论中出现的特定矩阵。进一步地，我们展示了如何利用该算法生成的矩阵，在每个存在相应联络的情形下，显式构造具有任意幂幺单值的Coxeter联络。