Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

TL;DR

This paper introduces algorithms for reducing unitary matrices to a CMV-like form, facilitating efficient eigenvalue computations and polynomial rootfinding, especially for rank-structured matrices.

Contribution

It presents a Lanczos-type and an elementary matrix-based algorithm for CMV-like reduction of unitary matrices, improving efficiency for rank-structured cases.

Findings

Algorithms achieve reduced complexity for rank-structured matrices.

The reduction enables fast eigenvalue computations using QR iteration.

Application to polynomial rootfinding demonstrates practical effectiveness.

Abstract

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos--type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+U^H. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank--structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank--structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration.