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.

翻译：通常需要构造对特定内积正交的多项式或有理函数。在实际数值算法中，这些多项式并非显式构造，而是计算相应的递推关系。此外，内积通常会被离散化为离散内积，即特定节点上加权函数求和的有限形式。对于特定应用而言，当在内积中增删节点时，若能高效更新递推关系将极具价值。递推关系的构造等价于计算具有指定谱性质的结构化矩阵（多项式情形）或矩阵束（有理函数情形），因此该问题的求解常被称为逆特征值问题。在Van Buggenhout等人（2022）的研究中，我们提出了在添加内积节点时高效更新递推关系的技术。为完善此项研究，本文提出了在内积中移除节点时高效降阶更新递推关系的方法。我们利用节点移除与QR算法中特征值收缩之间的内在联系开发高效算法，基于完美移位策略分别设计了多项式情形和有理函数情形的算法。数值实验验证了本方法的有效性。