We revisit the notion of root polynomials, thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020] for general polynomial matrices, and show how they can efficiently be computed in the case of matrix pencils. The staircase algorithm implicitly computes so-called zero directions, as defined in [P. Van Dooren, Computation of zero directions of transfer functions, Proceedings IEEE 32nd CDC, 3132--3137, 1993]. However, zero directions generally do not provide the correct information on partial multiplicities and minimal indices. These indices are instead provided by two special cases of zero directions, namely, root polynomials and vectors of a minimal basis of the pencil. We show how to extract, starting from the block triangular pencil that the staircase algorithm computes, both a minimal basis and a maximal set of root polynomials in an efficient manner. Moreover, we argue that the accuracy of the computation of the root polynomials can be improved by making use of iterative refinement.

翻译：我们重新审视了普通多元基质的根多面体概念,在[F. Dopico和V. Noferini, 根多面体及其在矩阵多面体理论中的作用,Linear Algebra Appl. 584:37-78, 2020] 中对普通多元基质的根多面体概念进行了透彻研究,并展示了如何在矩阵铅笔的情况下有效计算。楼梯算法暗含地计算了所谓的零方向,正如[P. Van Dooren, 计算转移函数零方向,Concess IEEEE 32nd CDC, 3132-3137, 1993] 所定义的。然而,一般而言,零方向并不提供部分多面和最低指数的正确信息。这些指数是由两个零方向的特殊例子提供的,即根多面体和铅笔最小基的矢量。我们展示了如何从块三角铅笔中提取出一个最小的基础和顶面的顶面多面值算法,既可以以有效的方式计算,也可以用一个顶点的顶部多面多面体精度模型进行精确的精确的计算。此外,我们论证是用多层的精度计算。