Implicit QR for Companion-like Pencils

TL;DR

This paper introduces a fast implicit QR algorithm tailored for eigenvalue problems involving low rank corrections of unitary matrices, specifically applied to polynomial zero-finding, achieving efficiency and stability.

Contribution

It presents a novel modified QZ algorithm that computes eigenvalues of structured matrix pencils with reduced computational complexity and memory usage.

Findings

Algorithm computes eigenvalues in O(N^2) operations

Numerical experiments confirm effectiveness and stability

Method is suitable for polynomial zero-finding problems

Abstract

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems . The modified QZ algorithm computes the generalized eigenvalues of certain NxN rank structured matrix pencils using O(N^2) ops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.