Structured conditioning of Hamiltonian eigenvalue problems

TL;DR

This paper analyzes how structure-preserving perturbations influence Hamiltonian eigenproblems, deriving formulas for condition numbers and worst-case effects, with special cases for purely imaginary eigenvalues.

Contribution

It provides new expressions for structured condition numbers and characterizes worst-case structured perturbations in Hamiltonian eigenproblems.

Findings

Structured perturbations significantly affect Hamiltonian eigenvalues.

Unstructured analysis suffices for purely imaginary eigenvalues.

Derived explicit formulas for structured condition numbers.

Abstract

We discuss the effect of structure-preserving perturbations on complex or real Hamiltonian eigenproblems and characterize the structured worst-case effect perturbations. We derive significant expressions for both the structured condition numbers and the worst-case effect Hamiltonian perturbations. It is shown that, for purely imaginary eigenvalues, the usual unstructured perturbation analysis is sufficient.