Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems

TL;DR

This paper introduces structure-preserving deflation techniques for critical eigenvalues in quadratic eigenvalue problems related to damped mass-spring systems, enhancing stability analysis and numerical computations.

Contribution

It proposes a novel structure-preserving deflation method using a trimmed linearization for quadratic eigenvalue problems in damped systems.

Findings

Effective deflation of eigenvalues at infinity, zero, and on the imaginary axis.

Application of methods to hyperbolic problems.

Analysis of damping matrix effects on eigenvalues.

Abstract

For a quadratic matrix polynomial associated with a damped mass-spring system there are three types of critical eigenvalues, the eigenvalues $\infty$ and $0$ and the eigenvalues on the imaginary axis. All these are on the boundary of the set of (robustly) stable eigenvalues. For numerical methods, but also for (robust) stability analysis, it is desirable to deflate such eigenvalues by projecting the matrix polynomial to a lower dimensional subspace before computing the other eigenvalues and eigenvectors. We describe structure-preserving deflation strategies that deflate these eigenvalues via a trimmed structure-preserving linearization. We employ these results for the special case of hyperbolic problems. We also analyze the effect of a (possibly low rank) parametric damping matrix on purely imaginary eigenvalues.