Stabilization of adaptive eigenvector extraction by continuation in nested orthogonal complement structure

Nguyen and Yamada [NY'13] proposed an adaptive algorithm for fast and stable extraction of the first generalized Hermitian eigenvector and mentioned the extension to the first r generalized eigenvector extraction based on the nested orthogonal complement structure [NTY'12]. However, we recently found that the estimates of the eigenvectors are not expressed ideally in the time-varying coordinate system and can change drastically in a certain situation, which may cause numerical instability. In this paper, we propose a new expression of the estimates along with time-varying coordinate system. This modification can be done efficiently with additional multiplications of orthogonal complement matrices. Numerical experiments show that the modified scheme has better stability compared with the original scheme [NTY'12].