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...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.展开更多
基金Partially Supported by JSPS KAKENHI(Grant No.JP16K05270)the Research Institute for Mathematical Sciences,a Joint Usage/Research Center located in Kyoto University
文摘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.
