SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1148-1171, January 2021.
We consider the computation of roots of polynomials expressed in the Chebyshev basis. We extend the QR iteration presented in [Y. Eidelman, L. Gemignani, and I. Gohberg, Numer. Algorithms, 47 (2008), pp. 253--273] introducing an aggressive early deflation strategy, and showing that the rank-structure allows one to parallelize the algorithm, avoiding data dependencies which would be present in the unstructured QR. We exploit the particular structure of the colleague linearization to achieve quadratic complexity and linear storage requirements. The (unbalanced) QR iteration used for Chebyshev rootfinding does not guarantee backward stability on the polynomial coefficients, unless the vector of coefficients satisfy $\norm{p} \approx 1$, a hypothesis which is almost never verified for polynomials approximating smooth functions. Even though the presented method is mathematically equivalent to the unbalanced QR algorithm, we show that exploiting the rank structure allows one to guarantee a small backward error on the polynomial, up to an explicitly computable amplification factor $\hat\gamma_1(p)$, which depends on the polynomial under consideration. We show that this parameter is almost always of moderate size, making the method accurate on several numerical tests, in contrast with what happens in the unstructured unbalanced QR.

中文翻译：

---

Chebyshev 求根的秩结构 QR

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 3 期，第 1148-1171 页，2021 年 1 月。
我们考虑以切比雪夫基表示的多项式根的计算。我们扩展了 [Y. Eidelman、L. Gemignani 和 I. Gohberg、Numer。Algorithms, 47 (2008), pp. 253--273] 介绍了一种积极的早期通缩策略，并表明秩结构允许并行化算法，避免非结构化 QR 中可能存在的数据依赖性。我们利用同事线性化的特殊结构来实现二次复杂度和线性存储要求。用于 Chebyshev 求根的（不平衡）QR 迭代不保证多项式系数的向后稳定性，除非系数向量满足 $\norm{p} \approx 1$，对于近似平滑函数的多项式几乎从未验证过这个假设。尽管所提出的方法在数学上等同于不平衡 QR 算法，但我们表明，利用秩结构可以保证多项式上的小后向误差，直至显式可计算的放大因子 $\hat\gamma_1(p)$，这取决于所考虑的多项式。我们表明，与非结构化不平衡 QR 中发生的情况相比，该参数几乎总是中等大小，使该方法在多个数值测试中准确无误。