On Computing the Kronecker Structure of Polynomial and Rational Matrices using Julia

TL;DR

This paper presents Julia-based algorithms for computing the Kronecker structure of polynomial and rational matrices, enabling analysis of spectral and singular properties crucial for solving related mathematical problems.

Contribution

It introduces reliable numerical algorithms for determining the Kronecker structure of linear matrix pencils within the Julia language, enhancing analysis of polynomial and rational matrices.

Findings

Effective algorithms for Kronecker structure computation

Implementation in Julia enhances computational efficiency

Illustrative examples demonstrate tool capabilities

Abstract

In this paper we discuss the mathematical background and the computational aspects which underly the implementation of a collection of Julia functions in the MatrixPencils package for the determination of structural properties of polynomial and rational matrices. We primarily focus on the computation of the finite and infinite spectral structures (e.g., eigenvalues, zeros, poles) as well as the left and right singular structures (e.g., Kronecker indices), which play a fundamental role in the structure of the solution of many problems involving polynomial and rational matrices. The basic analysis tool is the determination of the Kronecker structure of linear matrix pencils using numerically reliable algorithms, which is used in conjunction with several linearization techniques of polynomial and rational matrices. Examples of polynomial and rational matrices, which exhibit all relevant structural features, are considered to illustrate the main mathematical concepts and the capabilities of implemented tools.