论文标题
对多项式和一些合理的Krylov子空间的Chebyshev-Markov-Stieltjes的分离定理的综述
A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces
论文作者
论文摘要
高斯正交公式的累积正交权重在正交节点之间的间隔上构成了整体上的边界。此关注的经典结果可以追溯到Chebyshev，Markov和Stieltjes的作品，并被称为Chebyshev-Markov-Stieltjes（CMS定理）的分离定理。对于某些类别的理性高斯正交定理，类似的分离定理也是如此。给定矩阵和初始载体的Krylov子空间与给定基质的特征词中初始载体的光谱分布以及与此光谱分布相关的Riemann-Stielts的正交多项式密切相关。相似的关系对于理性的Krylov子空间也是如此。在目前的工作中，在Krylov子空间的背景下进行了分离定理，其中包括具有较高多重性和一些扩展Krylov子空间的单个复杂极的理性Krylov子空间。对于与具有单极的某些类别的理性Krylov子空间相关的理性高斯正交正交，因此在此新介绍了基本的分离定理。
The accumulated quadrature weights of Gaussian quadrature formulae constitute bounds on the integral over the intervals between the quadrature nodes. Classical results in this concern date back to works of Chebyshev, Markov and Stieltjes and are referred to as Separation Theorem of Chebyshev-Markov-Stieltjes (CMS Theorem). Similar separation theorems hold true for some classes of rational Gaussian quadrature. The Krylov subspace for a given matrix and initial vector is closely related to orthogonal polynomials associated with the spectral distribution of the initial vector in the eigenbasis of the given matrix, and Gaussian quadrature for the Riemann-Stielthes integral associated with this spectral distribution. Similar relations hold true for rational Krylov subspaces. In the present work, separation theorems are reviewed in the context of Krylov subspaces including rational Krylov subspaces with a single complex pole of higher multiplicity and some extended Krylov subspaces. For rational Gaussian quadrature related to some classes of rational Krylov subspaces with a single pole, the underlying separation theorems are newly introduced here.
