Geodesic
On the computation of the distance to quadratic matrix polynomials that are singular at some points on the unit circle
Electronic transactions on numerical analysis, Tome 42 (2014), pp. 165-176.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: For a quadratic matrix polynomial, the distance to the set of quadratic matrix polynomials which have singularities on the unit circle is computed using a bisection-based algorithm. The success of the algorithm depends on the eigenvalue method used within the bisection to detect the eigenvalues near the unit circle. To this end, the QZ algorithm along with the Laub trick is employed to compute the anti-triangular Schur form of a matrix resulting from a palindromic reduction of the quadratic matrix polynomial. It is shown that despite rounding errors, the Laub trick followed, if necessary, by a simple refinement procedure makes the results reliable for the intended purpose. Several numerical illustrations are reported.
Classification : 15A22, 65F35
Keywords: distance to instability, quadratic matrix polynomial, palindromic pencil, QZ algorithm, laub trick 
@article{ETNA_2014__42__a1,
     author = {Malyshev, Alexander and Sadkane, Miloud},
     title = {On the computation of the distance to quadratic matrix polynomials that are singular at some points on the unit circle},
     journal = {Electronic transactions on numerical analysis},
     pages = {165--176},
     publisher = {mathdoc},
     volume = {42},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2014__42__a1/}
}
TY  - JOUR
AU  - Malyshev, Alexander
AU  - Sadkane, Miloud
TI  - On the computation of the distance to quadratic matrix polynomials that are singular at some points on the unit circle
JO  - Electronic transactions on numerical analysis
PY  - 2014
SP  - 165
EP  - 176
VL  - 42
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ETNA_2014__42__a1/
LA  - en
ID  - ETNA_2014__42__a1
ER  - 
%0 Journal Article
%A Malyshev, Alexander
%A Sadkane, Miloud
%T On the computation of the distance to quadratic matrix polynomials that are singular at some points on the unit circle
%J Electronic transactions on numerical analysis
%D 2014
%P 165-176
%V 42
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ETNA_2014__42__a1/
%G en
%F ETNA_2014__42__a1
Malyshev, Alexander; Sadkane, Miloud. On the computation of the distance to quadratic matrix polynomials that are singular at some points on the unit circle. Electronic transactions on numerical analysis, Tome 42 (2014), pp. 165-176. http://geodesic.mathdoc.fr/item/ETNA_2014__42__a1/