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Verified Computation of Eigenpairs in the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils
作者姓名:Shinya  MIYAJIMA
作者单位:Faculty of Science and Engineering, Iwate University, Ueda, Morioka, 020-8551, Japan
基金项目:Partially Supported by JSPS KAKENHI (Grant No.JP16K05270) and the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
摘    要:Consider an optimization problem arising from the generalized eigenvalue problem Ax=λBx,where A,B∈Cm×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 CHC,where CH denotes the conjugate transpose of C,but cannot 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.

关 键 词:generalized  eigenvalue  problem  nonsquare  pencil  invariant  subspace  verified  numerical  computation
收稿时间:2019/5/7 0:00:00
修稿时间:2019/10/9 0:00:00

Verified Computation of Eigenpairs in the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils
Shinya MIYAJIMA.Verified Computation of Eigenpairs in the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils[J].Journal of Mathematical Research with Applications,2020,40(1):73-86.
Authors:Shinya MIYAJIMA
Institution:Faculty of Science and Engineering, Iwate University, Ueda, Morioka, 020-8551, Japan
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 cannot 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.
Keywords:generalized eigenvalue problem  nonsquare pencil  invariant subspace  verified numerical computation
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