Optimal Scaling of Generalized and Polynomial Eigenvalue Problems
Optimal Scaling of Generalized and Polynomial Eigenvalue Problems
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DOI：
10.1137/070704769
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发表时间：
2008-10
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影响因子：
--
通讯作者：
T. Betcke
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文献类型：
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作者：
T. Betcke
标度是标准特征值问题中常用的一种技术，可以提高特征值的灵敏度。本文研究任意次广义和多项式特征值问题的标度问题。证明了PEP相对于特征值的最优对角尺度可以用正态和分量条件数之比来描述。进一步研究了线性化对最优尺度多项式的影响。我们介绍了Lemonnier和Van Dooren对pep的对角缩放的推广，如果想得到的特征值的大小的一些信息是可用的，它是特别有效的，并且还讨论了任意程度pep的类型为$\lambda=\alpha\mu$的变量变换。
Scaling is a commonly used technique for standard eigenvalue problems to improve the sensitivity of the eigenvalues. In this paper we investigate scaling for generalized and polynomial eigenvalue problems (PEPs) of arbitrary degree. It is shown that an optimal diagonal scaling of a PEP with respect to an eigenvalue can be described by the ratio of its normwise and componentwise condition number. Furthermore, the effect of linearization on optimally scaled polynomials is investigated. We introduce a generalization of the diagonal scaling by Lemonnier and Van Dooren to PEPs that is especially effective if some information about the magnitude of the wanted eigenvalues is available and also discuss variable transformations of the type $\lambda=\alpha\mu$ for PEPs of arbitrary degree.