Quasiseparable Hessenberg reduction of real diagonal plus low rank matrices and applications

TL;DR

This paper introduces an efficient quasiseparable matrix-based algorithm for Hessenberg reduction of real diagonal plus low-rank matrices, improving computational cost and stability for polynomial eigenvalue problems.

Contribution

The paper presents a novel $O(n^2k)$ algorithm leveraging quasiseparable matrix technology for Hessenberg reduction of specific structured matrices.

Findings

Algorithm achieves $O(n^2k)$ complexity

Applications to polynomial eigenvalue problems demonstrated

Numerical experiments show stability of the method

Abstract

We present a novel algorithm to perform the Hessenberg reduction of an $n\times n$ matrix $A$ of the form $A = D + UV^*$ where $D$ is diagonal with real entries and $U$ and $V$ are $n\times k$ matrices with $k\le n$. The algorithm has a cost of $O(n^2k)$ arithmetic operations and is based on the quasiseparable matrix technology. Applications are shown to solving polynomial eigenvalue problems and some numerical experiments are reported in order to analyze the stability of the approach