The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems

In this paper, we investigate the convergence theory of generalized Lanczos methods for solving the eigenproblems of large unsymmetric matrices. Bounds for the distances between normalized eigenvectors and the Krylov subspace $\mathcal{K}_m (v_1 ,A)$ spanned by $v_1 , Av_1 , \ldots , A^{m - 1} v_1 $ are established, and a priori theoretical error bounds for eigenelements are presented when matrices are defective. Using them we show that the methods will still favor the outer part eigenvalues and the associated eigenvectors of A usually though they may converge quite slowly in the case of A being defective. Meanwhile, we analyze the relationships between the speed of convergence and the spectrum of A. However, a detailed analysis exposes that the approximate eigenvectors, Ritz vectors, obtained by generalized Lanczos methods for any unsymmetric matrix cannot be guaranteed to converge in theory even if approximate eigenvalues, Ritz values, do. Therefore, generalized Lanczos algorithms including Arnoldi’s algorithm and IOMs with correction are provided with necessary theoretical background.