A Symmetric Structure-Preserving GQR Algorithm for Linear Response Eigenvalue Problems

Tiexiang Li Ren-Cang Li Wen-Wei Lin

Rings and Algebras mathscidoc:1607.31001

In this paper, we present an efficient \GQR\ algorithm for solving the linear response eigenvalue problem $\scrH\bx ={\lambda}\bx $, where $\scrH$ is $\bPi^-$-symmetric with respect to $\Gamma_0 = \diag(I_n,-I_n)$. Based on newly introduced $\Gamma$-orthogonal transformations, the \GQR\ algorithm preserves the $\bPi^-$-symmetric structure of $\scrH$ throughout the whole process, which guarantees the computed eigenvalues to appear pairwise $(\lambda,-\lambda)$ as they should. With the help of a newly established implicit $\Gamma$-orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the \GQR\ algorithm. Numerical experiments are given to show the effectiveness of the algorithm.
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@inproceedings{tiexianga,
  title={A Symmetric Structure-Preserving GQR Algorithm for Linear Response Eigenvalue Problems},
  author={Tiexiang Li, Ren-Cang Li, and Wen-Wei Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160707121139452788045},
}
Tiexiang Li, Ren-Cang Li, and Wen-Wei Lin. A Symmetric Structure-Preserving GQR Algorithm for Linear Response Eigenvalue Problems. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160707121139452788045.
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