Generalized Hurwitz polynomials

Abstract

We describe a wide class of polynomials, which is a natural generalization of Hurwitz stable polynomials. We also give a detailed account of so-called self-interlacing polynomials, which are dual to Hurwitz stable polynomials but have only real and simple zeroes. All proofs are given using properties of rational functions mapping the upper half-plane of the complex plane to the lower half-plane. Matrices with self-interlacing spectra and other applications of generalized Hurwitz polynomials are discussed.

Hurwitz rational functions

“…We recall that n = r + m is the order of the rational function R. Now note that the formulae (1.8)-(1.9) are formal, so following verbatim the proof of the formulae (1.8)-(1.9) (see, for instance, [7,4]) one can establish the following relationships 1) for n = 2l,…”

“…For a given infinite sequence (s j ) ∞ j=0 , consider the determinants It is very well known [5,3] (see also [7]) that there are relations between the determinants D j (Φ), D j (Φ) and the Hurwitz minors ∆ j (p):…”

“…The following criterion of Hurwitz stability of a real polynomial was (implicitly) established in [5] (see also [3,1,7]).…”

“…It is well known that the problem of stability of a linear difference or differential system with constant coefficients reduces to the question of locating the zeroes of its characteristic polynomial in the left halfplane of the complex plane. One of the most famous results from stability theory is the Hurwitz theorem, which expresses stability of a real polynomial in terms of its coefficients [5,6,3] (see also [1,7]). Namely, the Hurwitz theorem states that a real polynomial has all its zeroes in the open left half-plane if and only if some determinants constructed with the coefficients of the polynomial are positive (see Theorem 1.5).…”

On the spectra of Schwarz matrices with certain sign patterns

“…The case of all b k negative was considered by O. Holtz [12]. Here we use formulae obtained by Wall in [17] which connect the entries of the matrix (1.1) with coefficients of its characteristic polynomial (see formulae (2.8) below) to use the so-called generalized Hurwitz theorem established in [16].…”