Several properties of invariant pairs of nonlinear algebraic eigenvalue problems
Several properties of invariant pairs of nonlinear algebraic eigenvalue problems
复制标题
非线性代数特征值问题不变对的几个性质
DOI：
10.1093/imanum/drt026
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发表时间：
2014
影响因子：
2.1
通讯作者：
Fei Xue
中科院分区：
文献类型：
--
作者：
D. Szyld;Fei Xue
本文分析了T（λ）v = 0形式的非线性代数特征值问题的不变对的几个重要性质。不变对子是与方阵的块Rayleigh行列式相关的不变子空间到非线性矩阵值函数T（·）的推广。它们在非线性特征值问题的分析和算法中起着重要的作用。本文首先证明了T（λ）v = 0的特征值所对应的代数重数、部分重数和几何重数以及Jordan链都可以完全由捕捉该特征值的简单不变对的Jordan标准形表示。然后，我们调查近似误差和扰动的一个简单的不变对。我们还表明，二阶精度的特征值近似可以实现双边块瑞利功能的非亏损特征值。最后，我们研究了特征值问题的Frechet导数的矩阵表示，并讨论了逆导数的范数估计，它度量了简单不变对的条件性和灵敏度。
We analyze several important properties of invariant pairs of nonlinear algebraic eigenvalue problems of the form T (λ)v = 0. Invariant pairs are generalizations of invariant subspaces in association with block Rayleigh quotients of square matrices to a nonlinear matrix-valued function T (·). They play an important role in the analysis of nonlinear eigenvalue problems and algorithms. In this paper, we first show that the algebraic, partial, and geometric multiplicities together with the Jordan chains corresponding to an eigenvalue of T (λ)v = 0 are completely represented by the Jordan canonical form of a simple invariant pair that captures this eigenvalue. We then investigate approximation errors and perturbations of a simple invariant pair. We also show that second order accuracy in eigenvalue approximation can be achieved by the two-sided block Rayleigh functional for non-defective eigenvalues. Finally, we study the matrix representation of the Frechet derivative of the eigenproblem, and we discuss the norm estimate of the inverse derivative, which measures the conditioning and sensitivity of simple invariant pairs.
影响因子：
1.1
作者：
Betcke T
通讯作者：
Betcke T