A new method for accelerating Arnoldi algorithms for large scale eigenproblems
Mathematics and Computers in Simulation
A refined jacobi-davidson method and its correction equation
Computers & Mathematics with Applications
Journal of Computational and Applied Mathematics
Further Analysis of the Arnoldi Process for Eigenvalue Problems
SIAM Journal on Numerical Analysis
Mathematical and Computer Modelling: An International Journal
Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem
Journal of Computational and Applied Mathematics
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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 ${\cal K}_m(v_1,A)$ 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.