Verified computation of eigenpairs in the generalized eigenvalue problem for nonsquare matrix pencils
Received:May 17, 2019  Revised:August 21, 2019
Key Word: Generalized eigenvalue problem, Nonsquare pencil, Invariant subspace, Verified numerical computation  
Fund ProjectL:JSPS KAKENHI
Author NameAffiliationE-mail
Shinya Miyajima Iwate University miyajima@iwate-u.ac.jp 
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Abstract:
      Consider an optimization problem arising from the generalized eigenvalue problem $Ax = \lambda Bx$, where $A,B \in \mathbb{C}^{m \times n}$ and $m > n$. Ito et al. showed that the optimization problem can be solved by utilizing right singular vectors of $C := [B,A]$. In this paper, we focus on computing intervals containing the solution. When some singular values of $C$ are multiple or nearly multiple, we can enclose bases of corresponding invariant subspaces of $C^HC$, where $C^H$ denotes the conjugate transpose of $C$, but can not enclose the corresponding right singular vectors. The purpose of this paper is to prove that the solution can be obtained even when we utilize the bases instead of the right singular vectors. Based on the proved result, we propose an algorithm for computing the intervals. Numerical results show property of the algorithm.
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