Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations

Authors:
D. Steven Mackey;Niloufer Mackey;Christian Mehl;Volker Mehrmann
Affiliations:
-;-;-;-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2006

Citing 0
Cited 17

Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms

Journal of Computational and Applied Mathematics
Critical delays and polynomial eigenvalue problems

Journal of Computational and Applied Mathematics
Singular-value-like decomposition for complex matrix triples

Journal of Computational and Applied Mathematics
A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems

Journal of Computational and Applied Mathematics
Structured Eigenvalue Condition Number and Backward Error of a Class of Polynomial Eigenvalue Problems

SIAM Journal on Matrix Analysis and Applications
Fiedler Companion Linearizations and the Recovery of Minimal Indices

SIAM Journal on Matrix Analysis and Applications
The Matrix Equation $X+A^TX^{-1}A=Q$ and Its Application in Nano Research

SIAM Journal on Scientific Computing
Palindromic companion forms for matrix polynomials of odd degree

Journal of Computational and Applied Mathematics
Solving a Structured Quadratic Eigenvalue Problem by a Structure-Preserving Doubling Algorithm

SIAM Journal on Matrix Analysis and Applications
Solving Rational Eigenvalue Problems via Linearization

SIAM Journal on Matrix Analysis and Applications
Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research

Journal of Computational and Applied Mathematics
Modifications of Newton's method for even-grade palindromic polynomials and other twined polynomials

Numerical Algorithms
NLEVP: A Collection of Nonlinear Eigenvalue Problems

ACM Transactions on Mathematical Software (TOMS)
Numerical studies on structure-preserving algorithms for surface acoustic wave simulations

Journal of Computational and Applied Mathematics
An algorithm for the complete solution of quadratic eigenvalue problems

ACM Transactions on Mathematical Software (TOMS)
Using permuted graph bases in H∞ control

Automatica (Journal of IFAC)
A closed-form estimator for the multivariate GARCH(1,1) model

Journal of Multivariate Analysis

Quantified Score

Hi-index	0.01

Visualization

Abstract

Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomials (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of linearizations which reflect the structure of these polynomials, and therefore preserve symmetries in their spectra, is introduced and investigated. We analyze the existence and uniqueness of such linearizations and show how they may be systematically constructed.