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.
翻译:Gaussian 二次方形的累积二次曲线重量是二次节点间隔间间隔的界限。 这一关注的经典结果可追溯到Chebyshev、Markov和Stieltjes的作品,称为Chebyshev-Markov-Stieltjes(CMS Theorem)的作品,称为Chebyshev-Markov-Stieltjes(CMS Theorem)的分离理论。类似的分离定理对于某些理性的高斯二次方形具有真实性。 Krylov 用于给定的矩阵和初始矢量的子空间的 Krylov 子空间子空间与特定矩阵的光谱分布相关,与给给定矩阵表的光谱分布相关,以及给Riemann-Stielts的立方形。对于理性的Krylov 子空间也存在相似的关系。在目前的工作中,对Krylov 子空间的分离进行了审查,包括理性的Krylov 亚方形子空间,与某种较复杂多的复合极和一些Krylov 基空间的扩展基层相联。