实线上离散多重正交多项式递推关系的更新与下推
Updating and Downdating the Recurrences of Discrete Multiple Orthogonal Polynomials on the Real Line
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中文总结 AI 辅助
该研究针对实线上离散多重正交多项式，提出基于逆特征值问题的递推矩阵更新方法与基于QR型算法启发的特征值压缩的下推方法，经数值实验验证了方法的稳定性与效率。
中文摘要 AI 辅助
我们研究实线上由多个正离散测度定义正交性的多重正交多项式，聚焦于I型和II型多重正交多项式及其步长递推关系，考虑从给定测度的节点和权重重构相关的带状上Hessenberg递推矩阵。我们提出高效的更新与下推程序，用于在离散测度中添加或移除节点时修改现有递推矩阵：更新策略基于求解逆特征值问题，下推程序则依赖受QR型算法启发的特征值压缩技术。数值实验证实了所提方法的稳定性与效率。
英文摘要
We study multiple orthogonal polynomials with orthogonality defined by multiple positive discrete measures on the real line. Focusing on type~I and type~II multiple orthogonal polynomials and their step-line recurrence relations, we consider the reconstruction of the associated banded upper Hessenberg recurrence matrix from given nodes and weights of the measures. We propose efficient updating and downdating procedures that modify an existing recurrence matrix when nodes are added to or removed from the discrete measures. The updating strategy is based on solving an inverse eigenvalue problem, while the downdating procedure relies on an eigenvalue deflation technique inspired by a QR-type of algorithm. Numerical experiments confirm the stability and efficiency of the proposed approaches.
