A Real QZ Algorithm for Structured Companion Pencils

TL;DR

This paper introduces a fast, stable, and memory-efficient implicit real QZ algorithm tailored for structured companion pencils, significantly improving eigenvalue computations in polynomial rootfinding.

Contribution

The paper presents a novel implicit real QZ algorithm optimized for structured companion pencils, reducing computational complexity and enhancing stability compared to existing methods.

Findings

Computes eigenvalues with O(N) flops per iteration

Uses O(N) memory storage

Demonstrates effectiveness through numerical experiments

Abstract

We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an $N\times N$ structured matrix pencil using $O(N)$ flops per iteration and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.