Fast QR iterations for unitary plus low rank matrices

TL;DR

This paper extends fast QR iteration algorithms to a broader class of matrices, specifically those that are a sum of a unitary matrix and a low-rank update, enabling efficient eigenvalue computations.

Contribution

The authors generalize existing algorithms to handle Hessenberg matrices with a unitary plus low-rank structure where the unitary part is not necessarily block circulant.

Findings

The extended approach maintains structural properties through a larger embedded matrix.

The factorization of the embedded matrix enables fast QR/QZ eigensolvers.

The algorithm is demonstrated to be fast and backward stable.

Abstract

Some fast algorithms for computing the eigenvalues of a block companion matrix $A = U + XY^H$, where $U\in \mathbb C^{n\times n}$ is unitary block circulant and $X, Y \in\mathbb{C}^{n \times k}$, have recently appeared in the literature. Most of these algorithms rely on the decomposition of $A$ as product of scalar companion matrices which turns into a factored representation of the Hessenberg reduction of $A$. In this paper we generalize the approach to encompass Hessenberg matrices of the form $A=U + XY^H$ where $U$ is a general unitary matrix. A remarkable case is $U$ unitary diagonal which makes possible to deal with interpolation techniques for rootfinding problems and nonlinear eigenvalue problems. Our extension exploits the properties of a larger matrix $\hat A$ obtained by a certain embedding of the Hessenberg reduction of $A$ suitable to maintain its structural properties. We show that $\hat A$ can be factored as product of lower and upper unitary Hessenberg matrices possibly perturbed in the first $k$ rows, and, moreover, such a data-sparse representation is well suited for the design of fast eigensolvers based on the QR/QZ iteration. The resulting algorithm is fast and backward stable.