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A Quadratic Eigenproblem in the Analysis of a Time Delay System
Author(s) -
Jarlebring Elias
Publication year - 2006
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200610017
Subject(s) - eigenvalues and eigenvectors , mathematics , quadratic equation , matrix (chemical analysis) , hermitian matrix , solver , divide and conquer eigenvalue algorithm , isotropic quadratic form , matrix differential equation , quadratic function , mathematical analysis , binary quadratic form , mathematical optimization , pure mathematics , geometry , differential equation , physics , materials science , quantum mechanics , composite material
Abstract In this work we solve a quadratic eigenvalue problem occurring in a method to compute the set of delays of a linear time delay system (TDS) such that the system has an imaginary eigenvalue. The computationally dominating part of the method is to find all eigenvalues z of modulus one of the quadratic eigenvalue problemwhere φ 1 , …, φ m –1 ∈ ℝ are free parameters and u a vectorization of a Hermitian rank one matrix. Because of its origin in the vectorization of a Lyapunov type matrix equation , the quadratic eigenvalue problem is, even for moderate size problems, of very large size. We show one way to treat this problem by exploiting the Lyapunov type structure of the quadratic eigenvalue problem when constructing an iterative solver. More precisely, we show that the shift‐invert operation for the companion form of the quadratic eigenvalue problem can be efficiently computed by solving a Sylvester equation . The usefulness of this exploitation is demonstrated with an example. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)