An Algorithm for the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils by Minimal Perturbation Approach

Shinji Ito, Kazuo Murota

Abstract: We deal with the generalized eigenvalue problem $A{\bm x} = \lambda B {\bm x}$ for nonsquare matrix pencils $ A - \lambda B $, where $ A , B \in \mathbb{C}^{m \times n}$ and $m > n$. A major difficulty inherent in this problem is that perturbation to inputs may cause eigenvalues to fail to exist even if eigenvalues are known to exist in the noiseless case. To cope with this situation, Boutry et al. [SIAM J. Matrix Anal. Appl., 27 (2005), pp. 582--601] have proposed a novel approach that searches for the minimal perturbation to the pencil such that the perturbed pencil has eigenpairs. Boutry et al. first aimed at finding the minimal perturbation such that the perturbed pencil has $n$ eigenpairs, but they settled for a simplified version that guarantees at least one eigenpair. The aim of this paper is to present an algorithm for the original version of the problem with $n$ eigenpairs. The proposed algorithm is based on the total least squares problem introduced by Golub and Van Loan. The algorithm is much simpler and runs faster than Boutry et al.'s algorithm. It is confirmed numerically that the proposed algorithm is more robust against data noise than Boutry et al.'s algorithm.