An Elementary Derivation of the Projection Method for Nonlinear Eigenvalue Problems Based on Complex Contour Integration
An Elementary Derivation of the Projection Method for Nonlinear Eigenvalue Problems Based on Complex Contour Integration
复制标题
基于复轮廓积分的非线性特征值问题投影法的初等推导
DOI：
10.1007/978-3-319-62426-6_16
复制
发表时间：
2015
影响因子：
3
通讯作者：
Yusaku Yamamoto
中科院分区：
文献类型：
--
作者：
Yusaku Yamamoto
本文将求解广义特征值问题的Sakurai-Sugiura（SS）投影法推广到非线性特征值问题A（z）w = 0，其中A（z）是一个解析矩阵值函数。据作者所知，这些方法的现有推导依赖于解析矩阵函数的标准形式，例如史密斯形式或凯尔迪什定理。虽然这些定理是强大的工具，但它们需要分析和线性代数的高级知识，即使在线性代数的高级教科书中也很少提及。本文给出了求解非线性特征值问题的SS型算法的初等推导。我们的推导仅使用参数化矩阵A（z）的特征值和特征向量的解析性，这是矩阵扰动理论中的标准结果。因此，我们希望我们的方法将提供一个容易访问的路径的非线性SS型方法的理论。
The Sakurai-Sugiura (SS) projection method for the generalized eigenvalue problem has been extended to the nonlinear eigenvalue problem A(z)w = 0, where A(z) is an analytic matrix valued function, by several authors. To the best of the authors’ knowledge, existing derivations of these methods rely on canonical forms of an analytic matrix function such as the Smith form or the theorem of Keldysh. While these theorems are powerful tools, they require advanced knowledge of both analysis and linear algebra and are rarely mentioned even in advanced textbooks of linear algebra. In this paper, we present an elementary derivation of the SS-type algorithm for the nonlinear eigenvalue problem, assuming that the wanted eigenvalues are all simple. Our derivation uses only the analyticity of the eigenvalues and eigenvectors of a parametrized matrix A(z), which is a standard result in matrix perturbation theory. Thus we expect that our approach will provide an easily accessible path to the theory of nonlinear SS-type methods.