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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 017 902 888  
Author(s):  
Highham, Nicolas J.; Tisseur, Francoise:  
Shorttitle:  
More on pseudospectra for polynomial eigenvalue problems and applications in control theory  
Source:  
Linear Algebra Appl. 351352, 435  453 (2002).  
Classification:  
 
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
polynomial eigenvalue problem; pseudospectrum; nearest nonregular polynomial; nearest uncontrollable dynamical system; structured perturbations  
Review:  
Next to the usual linear spectral (or eigenvalue) problem A x = e x one finds the generalized one, A x = e B x. Moving on, a squarematrix polynomial problem follows, P(e) x = 0. Its nonsquare form appeals finally to the authors motivated (say, by the theory of games) to study the ``infinite" eigenvalues e = a/b with b = 0. They contemplate the homogeneous polynomial and add perturbations, [P(a,b)+ D P'(a,b)] x = 0. In this way the authors define the (complex) pseudospectrum [= a set of pairs (a,b)] and show how it can be calculated via minimization of P x over x with unit norm (cf. their main Theorem 2.1). In the sequel they illustrate the concept (on the Wilkinson's pathological example), suggest a vizualization on the Riemann sphere (a projection which treats also the information about the infinite eigenvalues properly) and outline a few applications [studying the distance from the nearest nonregular matrix polynomial or from a domain of uncontrollability (= impossibility to reach some final states) in a dynamical system]. Finally they extend the concept, in the spirit of their preceding paper [21], to the so called structured pseudospectra and illustrate their merits on a damped massspring example of the Tisseur's older paper [20].  
Remarks to the editors:  
I do not mind replacements of my plaintext equations by their improved TEX forms (then, you could also replace e by lambda, a by \alpha, b by \beta and D by capital \Delta).  