In this article, we describe the theoretical foundations of the Macaulay2 package ConnectionMatrices and explain how to use it. For a left ideal in the Weyl algebra that is of finite holonomic rank, we implement the computation of the encoded system of linear PDEs in connection form with respect to an elimination term order that depends on a chosen positive weight vector. We also implement the gauge transformation for carrying out a change of basis over the field of rational functions. We demonstrate all implemented algorithms with examples.

翻译：本文阐述了Macaulay2软件包ConnectionMatrices的理论基础并说明其使用方法。针对具有有限正则秩的Weyl代数左理想，我们实现了以连接形式编码的线性偏微分方程组的计算，该计算基于取决于选定正权向量的消元项序。我们还实现了在有理函数域上执行基变换的规范变换。通过具体示例展示了所有已实现算法的应用。