同事矩阵对角化的可证逐点向后稳定$O(n^2)$ QR算法
A Provably Componentwise Backward Stable $O(n^2)$ QR Algorithm for the Diagonalization of Colleague Matrices
摘要：
已知在切比雪夫基中表示的首一多项式的根是所谓的同事矩阵的特征值，同事矩阵是对称三对角矩阵和秩1矩阵之和的Hessenberg矩阵。因此，求根问题被重新表述为一个特征问题，使得这种矩阵的特征值的计算具有重要的实际意义。在这篇手稿中，我们描述了一个$O(n^2)$显式结构QR算法的同事矩阵，并证明了它是分量向后稳定的，在这个意义上，同事矩阵的向后误差可以表示为相对扰动其组件。Noferini、Robol和Vandebril最近的一个结果表明，分量向后稳定性意味着多项式的切比雪夫展开系数的向量$c$中的向后误差$δc$有界$\lVert δc \rVert \lesssim \lVert c \rVert u$，其中$u$是机器精度。因此，我们所描述的算法具有系数中的最优向后误差和最优成本$O(n^2)$。我们说明了几个数值例子的算法的性能。
英文摘要：
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 $δc$ in the vector $c$ of Chebyshev expansion coefficients of the polynomial has the bound $\lVert δc \rVert \lesssim \lVert c \rVert u$, 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.
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