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.

中文翻译：

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同事矩阵对角线化的逐分量向后稳定$ O（n ^ 2）$ QR算法

以Chebyshev为基础表示的单项多项式的根已知是所谓的同事矩阵的特征值，该同事矩阵是Hessenberg矩阵，它是对称三对角矩阵和rank-1矩阵的和。寻根问题因此被重新表述为一个本征问题，从而使此类矩阵的本征值的计算成为具有重大实际意义的主题。在此手稿中，我们描述了一种针对同事矩阵的$ O（n ^ 2）$显式结构化QR算法，并证明了它是分量向后稳定的，因为可以将同事矩阵中的向后误差表示为其对其的相对扰动成分。Noferini，Robol的最新结果，Vandebril表示，逐项向后稳定性意味着多项式Chebyshev展开系数的向量$ c $中的向后误差$ \ delta c $具有绑定的\\ lVert \ delta c \ rVert \ lesssim \ lVert c \ rVert u $ ，其中$ u $是机器精度。因此，我们描述的算法既具有最佳的系数后向误差，又具有最佳的成本$ O（n ^ 2）$。我们通过几个数值示例来说明该算法的性能。