Naslov
Singular quadratic eigenvalue problems: Linearization and weak condition numbers

Autori
Kressner, Daniel ; Šain Glibić, Ivana

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, neobjavljeni rad, znanstveni

Skup
3rd Workshop on Optimal Control of Dynamical Systems and applications

Mjesto i datum
Osijek, Hrvatska, 28.03.2022. - 01.04.2022

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
Singular eigenvalue problems ; Polynomial eigenvalue problem ; Linearization ; Weak condition number

Sažetak
In this talk we will study the numerical computation of singular quadratic eigenvalue problem associated with matrix polynomial $Q(\lambda) = \lambda^2M+\lambda C + K$ such that $\det(Q(\lambda))\equiv 0$. Since small matrix perturbation of singular problems can move eigenvalues anywhere in the complex plain these problems are known to be ill--conditioned. However, it was recognized by Wilkinson, and later proven by De Ter\'{; ; a}; ; n, Dopico and Moro that perturbation directions causing arbitrary large eigenvalue changes are rare. More recently, in order to quantify the eigenvalue sensitivity, Lotz and Noferini considered perturbation directions which are uniformly distributed on the sphere. They introduced the so called $\delta$--weak condition number that bounds the eigenvalue sensitivity with probability $1- \delta$, $0\leq\delta<1$, together with an upper bound. Since linearization is logical first step for the solution of polynomial eigenvalue problems, we supplement the theory of $\delta$--weak condition number with its lower bound in order to prove that one can always choose the linearization for this problem so that this quantity is not affected. In addition, we develop theoretical criterion for classification of eigenvalues of a perturbed matrix polynomial leading to a new procedure for the solution of singular quadratic eigenvalue problem. We prove that well--conditioned finite eigenvalues can be detected with high probability. Finally, we will present numerical experiments to demonstrate the efficiency of proposed method.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA