A bisection method for measuring the distance of a quadratic matrix polynomial to the quadratic matrix polynomials that are singular on the unit circle

Abstract

The computation of the distance of a quadratic matrix polynomial to the quadratic matrix polynomials that are singular on the unit circle is investigated. The emphasis is placed on backward stable methods that transform the computation of the distance to a palindromic eigenvalue problem for which structure-preserving eigensolvers can be utilized in conjunction with a bisection algorithm. Reliability of the suggested methods is guaranteed by a novel error analysis.

Appendix: Another proof of Byers’ results in [7, Thm 3]

Assume that the eigenvalues of the Hamiltonian matrix , parameterized by σ≥0, are computed by a structure-preserving backward stable numerical algorithm, i.e., the computed eigenvalues are the exact eigenvalues of a perturbed matrix H(σ)+Δ, where the perturbation Δ is Hamiltonian, that is, ΔJ is Hermitian for . Let us denote $r=\min_{\omega\in\mathbb{R}}\sigma_{\min}(A-i\omega I)$ and δ=∥Δ∥₂. Theorem 3 in [7] is essentially equivalent to the following two assertions:

• if one of the computed eigenvalues lies on the imaginary axis, then r≤σ+δ;

• if all the computed eigenvalues lie outside the imaginary axis, then r>σ−δ.

Below we provide a new proof, which uses the same technique as the arguments from Sect. 4.1.

First we observe that the eigenvalues of the Hermitian matrix

$$S(\omega)=H(\sigma)J-i\omega J= \begin{bmatrix}A-i\omega I&\sigma I\\ -\sigma I&-A^*-i\omega I\end{bmatrix}J= \begin{bmatrix}-\sigma I&A-i\omega I\\ (A-i\omega I)^*&-\sigma I\end{bmatrix} $$

are ±s _k (ω)−σ, k=1,…,n, where s _k (ω) are the singular values of A−iωI, $\omega\in\mathbb{R}$. It is easy to see that the set $\{\min_{k}s_{k}(\omega)\colon\omega\in\mathbb{R}\}$ equals the interval [r,∞), and the set $\{s_{k}(\omega)\colon\omega\in\mathbb{R}, k=1,\ldots,n\}$ coincides with [r,∞) too. Hence the set of all eigenvalues of S(ω) for all real ω equals (−∞,−r−σ]∪[r−σ,∞).

Owing to the Courant-Fischer minimax theorem [10] the set of all eigenvalues of the perturbed matrix S(ω)+ΔJ for all ω contains the interval [r+δ−σ,∞) and does not contain the interval (−r+δ−σ,r−δ−σ). When σ∈[r+δ,∞), the former interval contains 0, i.e., the perturbed Hamiltonian matrix H(σ)+Δ has an eigenvalue on the imaginary axis. When δ<r and σ∈[0,r−δ), the latter interval contains 0, and the perturbed Hamiltonian matrix H(σ)+Δ has no eigenvalues on the imaginary axis.