A Provably Componentwise Backward Stable $O(n^2)$ QR Algorithm for the Diagonalization of Colleague Matrices

TL;DR

This paper introduces an $O(n^2)$ structured QR algorithm for colleague matrices that is provably componentwise backward stable, enabling accurate and efficient polynomial rootfinding via eigenvalue computation.

Contribution

The paper presents the first explicit $O(n^2)$ QR algorithm for colleague matrices with proven componentwise backward stability, improving both accuracy and computational efficiency.

Findings

The algorithm achieves optimal backward error in Chebyshev coefficients.

Numerical examples demonstrate the algorithm's stability and efficiency.

The method outperforms previous approaches in stability and cost.

Abstract

The roots of a monic polynomial expressed in a Chebyshev basis are known to be the eigenvalues of the so-called colleague matrix, which is a Hessenberg matrix that is the sum of a symmetric tridiagonal matrix and a rank-1 matrix. The rootfinding problem is thus reformulated as an eigenproblem, making the computation of the eigenvalues of such matrices a subject of significant practical importance. In this manuscript, we describe an $O(n^2)$ explicit structured QR algorithm for colleague matrices and prove that it is componentwise backward stable, in the sense that the backward error in the colleague matrix can be represented as relative perturbations to its components. A recent result of Noferini, Robol, and Vandebril shows that componentwise backward stability implies that the backward error $\delta c$ in the vector $c$ of Chebyshev expansion coefficients of the polynomial has the bound $\lVert \delta c \rVert \lesssim \lVert c \rVert u$, where $u$ is machine precision. Thus, the algorithm we describe has both the optimal backward error in the coefficients and the optimal cost $O(n^2)$. We illustrate the performance of the algorithm with several numerical examples.