# Pole-swapping algorithms for alternating and palindromic eigenvalue problems

## TL;DR

This paper introduces structure-preserving pole-swapping algorithms for palindromic and alternating eigenvalue problems, ensuring backward stability and extending stability guarantees to previous bulge-chasing methods.

## Contribution

It develops new pole-swapping algorithms tailored for palindromic and alternating eigenvalue problems, with a refinement step to ensure backward stability.

## Key findings

- Algorithms are structure-preserving for specific eigenvalue problems.
- Refinement step guarantees backward stability.
- Backward stability is extended to existing bulge-chasing algorithms.

## Abstract

Pole-swapping algorithms are generalizations of bulge-chasing algorithms for the generalized eigenvalue problem. Structure-preserving pole-swapping algorithms for the palindromic and alternating eigenvalue problems, which arise in control theory, are derived. A refinement step that guarantees backward stability of the algorithms is included. This refinement can also be applied to bulge-chasing algorithms that had been introduced previously, thereby guaranteeing their backward stability in all cases.