# On pole-swapping algorithms for the eigenvalue problem

## TL;DR

This paper develops a new, flexible convergence theory for pole-swapping algorithms used in eigenvalue problems, introduces an improved swapping routine, and explores their implications for algorithm design.

## Contribution

A new modular convergence theory for pole-swapping algorithms and an improved swapping routine demonstrated through analysis and numerical tests.

## Key findings

- The new swapping routine outperforms existing methods.
- The modular theory applies broadly to all pole-swapping algorithms.
- Insights into bi-directional chasing and shift strategies are provided.

## Abstract

Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms.