The conditioning of block Kronecker $\ell$-ifications of matrix polynomials

TL;DR

This paper analyzes the eigenvalue conditioning of block Kronecker $ ext{ell}$-ifications of matrix polynomials, showing they are comparably well-conditioned to the original polynomial under certain scaling, and extends understanding of their numerical stability.

Contribution

It provides a detailed comparison of the conditioning of block Kronecker $ ext{ell}$-ifications with the original polynomial, establishing conditions for their numerical stability.

Findings

Block Kronecker $ ext{ell}$-ifications are as well-conditioned as the original polynomial under proper scaling.

Any block Kronecker companion form is comparably conditioned to Frobenius companion forms.

Numerical examples support the theoretical conditioning results.

Abstract

A strong $\ell$-ification of a matrix polynomial $P(\lambda)=\sum A_i\lambda^i$ of degree $d$ is a matrix polynomial $\mathcal{L}(\lambda)$ of degree $\ell$ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as $P(\lambda)$. Strong $\ell$-ifications can be used to transform the polynomial eigenvalue problem associated with $P(\lambda)$ into an equivalent polynomial eigenvalue problem associated with a larger matrix polynomial $\mathcal{L}(\lambda)$ of lower degree. Typically $\ell=1$ and, in this case, $\mathcal{L}(\lambda)$ receives the name of strong linearization. However, there exist some situations, e.g., the preservation of algebraic structures, in which it is more convenient to replace strong linearizations by other low degree matrix polynomials. In this work, we investigate the eigenvalue conditioning of $\ell$-ifications from a family of matrix polynomials recently identified and studied by Dopico, P\'erez and Van Dooren, the so-called block Kronecker companion forms. We compare the conditioning of these $\ell$-ifications with that of the matrix polynomial $P(\lambda)$, and show that they are about as well conditioned as the original polynomial, provided we scale $P(\lambda)$ so that $\max\{\|A_i\|_2\}=1$, and the quantity $\min\{\|A_0\|_2,\|A_d\|_2\}$ is not too small. Moreover, under the scaling assumption $\max\{\|A_i\|_2\}=1$, we show that any block Kronecker companion form, regardless of its degree or block structure, is about as well-conditioned as the well-known Frobenius companion forms. Our theory is illustrated by numerical examples.