改变离散内积时修改正交多项式与有理函数递推关系的算法
Algorithms for Modifying Recurrence Relations of Orthogonal Polynomial and Rational Functions when Changing the Discrete Inner Product
AI总结：
本文研究了在离散内积中移除节点时高效降维正交多项式与有理函数递推关系的算法，利用移除节点与QR算法收缩特征值的联系，基于完美位移策略开发了适用于多项式和有理函数的算法，并通过数值实验验证了方法的有效性。
摘要：
通常，我们需要针对特定内积正交的多项式或有理函数。在实际的数值算法中，这些多项式本身并不被构造，而是计算相关的递推关系。此外，内积通常也会被改变为离散内积，即在特定节点处求值的函数的有限加权和。对于特定的应用，拥有一个高效的程序来在内积中添加或移除节点时更新递推关系是有益的。递推关系的构造等价于计算具有预定谱性质的结构化矩阵（多项式情况）或矩阵束（有理函数情况）。因此，该问题的求解通常被称为求解逆特征值问题。在 Van Buggenhout et al. (2022) 中，我们提出了在向内积添加节点时高效更新递推关系的更新技术。为了完善这项研究，我们在本文中提出了在内积中移除节点时高效降维（downdate）递推关系的方法。我们利用了移除节点与 QR 算法收缩特征值之间的联系来开发高效算法。我们将基于完美位移策略，为多项式情况和有理函数环境分别开发算法。数值实验验证了我们的方法。
英文摘要：
Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed. Moreover, also typically the inner product is changed to a discrete inner product, which is the finite sum of weighted functions evaluated in specific nodes. For particular applications it is beneficial to have an efficient procedure to update the recurrence relations when adding or removing nodes from the inner product. The construction of the recurrence relations is equivalent to computing a structured matrix (polynomial) or pencil (rational) having prescribed spectral properties. Hence the solution of this problem is often referred to as solving an Inverse Eigenvalue Problem. In Van Buggenhout et al. (2022) we proposed updating techniques to add nodes to the inner product while efficiently updating the recurrences. To complete this study we present in this article manners to efficiently downdate the recurrences when removing nodes from the inner product. The link between removing nodes and the QR algorithm to deflate eigenvalues is exploited to develop efficient algorithms. We will base ourselves on the perfect shift strategy and develop algorithms, both for the polynomial case and the rational function setting. Numerical experiments validate our approach.
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