Computing eigenvalues of normal matrices via complex symmetric matrices

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:259

Article; Proceedings Paper

Ferranti, Micol¹  Vandebril, Raf¹ 

[1] Katholieke Univ Leuven, Dept Comp Sci, B-3000 Louvain, Belgium

关键词: Normal matrix;    Complex symmetric;    Takagi factorization;    Unitary similarity;    Symmetric singular value decomposition;    Eigenvalue decomposition;   

DOI  :  10.1016/j.cam.2013.08.036

来源: Elsevier

【 摘 要 】

Computing all eigenvalues of a modest size matrix typically proceeds in two phases. In the first phase, the matrix is transformed to a suitable condensed matrix format, sharing the eigenvalues, and in the second stage the eigenvalues of this condensed matrix are computed. The main purpose of this intermediate matrix is saving valuable computing time. Important subclasses of normal matrices, such as the Hermitian, skew-Hermitian and unitary matrices admit a condensed matrix represented by only O(n) parameters, allowing subsequent low-cost algorithms to compute their eigenvalues. Unfortunately, such a condensed format does not exist for a generic normal matrix. We will show, under modest constraints, that normal matrices also admit a memory cheap intermediate matrix of tridiagonal complex symmetric form. Moreover, we will propose a general approach for computing the eigenvalues of a normal matrix, exploiting thereby the normal complex symmetric structure. An analysis of the computational cost and numerical experiments with respect to the accuracy of the approach are enclosed. In the second part of the manuscript we will investigate the case of nonsimple singular values and propose a theoretical framework for retrieving the eigenvalues. We will, however, also highlight some numerical difficulties inherent to this approach. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free