A Provably Componentwise Backward Stable $O(n^2)$ QR Algorithm for the Diagonalization of Colleague Matrices
Abstract
The roots of a monic polynomial expressed in a Chebyshev basis are known to be the eigenvalues of the socalled colleague matrix, which is a Hessenberg matrix that is the sum of a symmetric tridiagonal matrix and a rank1 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.12186
 arXiv:
 arXiv:2102.12186
 Bibcode:
 2021arXiv210212186S
 Keywords:

 Mathematics  Numerical Analysis
 EPrint:
 51 pages, 10 figures