On computing root polynomials and minimal bases of matrix pencils

TL;DR

This paper revisits the concept of root polynomials for matrix pencils, proposing efficient methods to compute minimal bases and root polynomials, and enhancing accuracy through iterative refinement.

Contribution

It introduces an efficient approach to compute minimal bases and root polynomials from staircase algorithm outputs, improving accuracy with iterative refinement.

Findings

Efficient extraction of minimal bases and root polynomials from block triangular pencils.

Iterative refinement improves the accuracy of computed root polynomials.

Clarifies the distinction between zero directions and root polynomials in matrix pencil analysis.

Abstract

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.