Detecting hyperbolic and definite matrix polynomials

Publikationstyp
Journal Article
Date Issued
2009-11-17
Sprache
English
Author(s)
Niendorf, Vasco  
Voß, Heinrich 
Institut
Mathematik E-10  
Numerische Simulation E-10 (H)  
TORE-URI
http://hdl.handle.net/11420/3904
Journal
Linear algebra and its applications  
Volume
432
Issue
4
Start Page
1017
End Page
1035
Citation
Linear Algebra and Its Applications 4 (432): 1017-1035 (2010)
Publisher DOI
10.1016/j.laa.2009.10.014
Scopus ID
2-s2.0-71649103477
Publisher
American Elsevier Publ.
Hyperbolic or more generally definite matrix polynomials are important classes of Hermitian matrix polynomials. They allow for a definite linearization and can therefore be solved by a standard algorithm for Hermitian matrices. They have only real eigenvalues which can be characterized as minmax and maxmin values of Rayleigh functionals, but there is no easy way to test if a given polynomial is hyperbolic or definite or not. Taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil. Numerical examples demonstrate the efficiency of the approach.
Subjects
Definite matrix polynomial
Hyperbolic
Matrix polynomial
Minmax characterization
Overdamped
Quadratic eigenvalue problem
Safeguarded iteration
DDC Class
510: Mathematik