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The roots of a monic polynomial expressed in a Chebyshev basis are known tobe the eigenvalues of the so-called colleague matrix, which is a Hessenbergmatrix that is the sum of a symmetric tridiagonal matrix and a rank-1 matrix.The rootfinding problem is thus reformulated as an eigenproblem, making thecomputation of the eigenvalues of such matrices a subject of significantpractical importance. In this manuscript, we describe an $O(n^2)$ explicitstructured QR algorithm for colleague matrices and prove that it iscomponentwise backward stable, in the sense that the backward error in thecolleague matrix can be represented as relative perturbations to itscomponents. A recent result of Noferini, Robol, and Vandebril shows thatcomponentwise backward stability implies that the backward error $\delta c$ inthe vector $c$ of Chebyshev expansion coefficients of the polynomial has thebound $\lVert \delta c \rVert \lesssim \lVert c \rVert u$, where $u$ is machineprecision. Thus, the algorithm we describe has both the optimal backward errorin the coefficients and the optimal cost $O(n^2)$. We illustrate theperformance of the algorithm with several numerical examples.

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