Fast Hessenberg reduction of some rank structured matrices

TL;DR

This paper introduces two efficient algorithms for Hessenberg reduction of rank-structured matrices, significantly improving computational speed for matrices of the form D + UV^H and extending to unitary plus low-rank matrices.

Contribution

The paper presents novel, faster algorithms for Hessenberg reduction of structured matrices, including a two-stage approach and a block CMV-based method, with proven linear complexity in rank.

Findings

Achieves $O(n^2k)$ complexity for real matrices.

Extends reduction techniques to unitary plus low-rank matrices.

Provides a numerically stable, condensed representation of the reduced matrix.

Abstract

We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$ where $D$ is a real or unitary $n \times n$ diagonal matrix and $U, V \in\mathbb{C}^{n \times k}$. The proposed algorithm for the real case exploits a two--stage approach by first reducing the matrix to a generalized Hessenberg form and then completing the reduction by annihilation of the unwanted sub-diagonals. It is shown that the novel method requires $O(n^2k)$ arithmetic operations and it is significantly faster than other reduction algorithms for rank structured matrices. The method is then extended to the unitary plus low rank case by using a block analogue of the CMV form of unitary matrices. It is shown that a block Lanczos-type procedure for the block tridiagonalization of $\Re(D)$ induces a structured reduction on $A$ in a block staircase CMV--type shape. Then, we present a numerically stable method for performing this reduction using unitary transformations and we show how to generalize the sub-diagonal elimination to this shape, while still being able to provide a condensed representation for the reduced matrix. In this way the complexity still remains linear in $k$ and, moreover, the resulting algorithm can be adapted to deal efficiently with block companion matrices.