An adaptive extraction of generalized eigensubspace by using exact nested orthogonal complement structure

• DOI: 10.1007/s11045-012-0172-9
• Publication Date: 2012-01-20T06:21:39Z
• Authors: Tuan Duong Nguyen; Noriyuki Takahashi; Isao Yamada
• Keywords: Orthogonal complement; Minor (academic); Matrix pencil; Generalized eigenvector; Matrix (chemical analysis); Orthonormality; Pencil (optics)

The contribution of this paper is three-fold: first, we propose a novel scheme for generalized minor subspace extraction by extending an idea of dimension reduction technique. The key of this scheme is the reduction of the problem for extracting the ith (i ? 2) minor generalized eigenvector of the original matrix pencil to that for extracting the first minor generalized eigenvector of a matrix pencil of lower dimensionality. The proposed scheme can employ any algorithm capable of estimating the first minor generalized eigenvector. Second, we propose a pair of such iterative algorithms and analyze their convergence properties in the general case where the generalized eigenvalues are not necessarily distinct. Third, by using these algorithms inductively, we present adaptive implementations of the proposed scheme for estimating an orthonormal basis of the generalized minor subspace. Numerical examples show that the proposed adaptive subspace extraction algorithms have better numerical stability than conventional algorithms.