Definitions and characterizations of pseudospectra are given for rectangular matrix polynomials expressed in homogeneous form: P(α, β) = αAd + αd−1βAd−1 + · · · + βA0. It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann sphere is shown to provide a convenient way to visualize pseudospect...

The eigenproblems of the rank-one updates of the matrices have lots of applications in scientific computation and engineering such as the symmetric tridiagonal eigenproblems by the divide-andconquer method and Web search engine. Many researchers have well studied the algorithms for computing eigenvalues of Hermitian matrices updated by a rank-one matrix [1–6]. Recently, Ding and Zhou studied a ...

We show that 2-norm pseudospectra ofm-by-nmatrices have no more than 2m(4m−1) connected components. Such bounds are pertinent for computing the distance to uncontrollability of a control system, since this distance is the minimum value of a function whose level sets are pseudospectra. We also discuss algorithms for computing this distance, including a trisection variant of Gu’s recent algorithm...

The aim of this paper is to analyze the relative importance of the dimension and the structure of the square matrices in the quantification of their non-normality. We envisage non-normal matrices which come from numerical analysis of o. d. e. and p. d. e. as well as from various iterative processes where the parameter dimension varies. The main result consists in an upper bound for the departur...

In this paper we consider a linear wave equation with strong damping and dynamical boundary conditions as an alternative model for the classical spring-mass-damper ODE. Our purpose is to compare analytically these two approaches to the same physical system. We take a functional analysis point of view based on semigroup theory, spectral perturbation analysis and dominant eigenvalues.

We consider the possible detection of parity violation at the linear level in gravity using polarized anisotropies of the cosmic microwave background. Since such a parity violation would lead to nonzero TB and EB correlations, this makes those odd-parity angular power spectra a potential probe of parity violation in the gravitational sector. These spectra are modeled incorporating the impact of...

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct int...

Two useful measures of the robust stability of the discrete-time dynamical system xk+1 = Axk are the -pseudospectral radius and the numerical radius of A. The -pseudospectral radius of A is the largest of the moduli of the points in the -pseudospectrum of A, while the numerical radius is the largest of the moduli of the points in the field of values. We present globally convergent algorithms fo...

Non-Hermitian topological systems simultaneously possess two antagonistic features: ultrasensitivity due to exceptional points and robustness of zero-energy modes, it is unclear which one prevails under different perturbations. We study that question by applying the pseudospectrum theory on prototypical non-Hermitian Su-Schrieffer-Heeger lattice. Topological modes around underlying third-order ...