Abstract :
Arnoldiʹs method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices. However, the approximate eigenvectors or Ritz vectors obtained by Arnoldiʹs method cannot be guaranteed to converge in theory even if the approximate eigenvalues or Ritz values do. In order to circumvent this potential danger, a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions. It is shown that refined approximate eigenvectors converge to eigenvectors if Ritz values do. Moreover, the resulting refined algorithms converge more rapidly. We report some numerical experiments and compare the refined algorithms with their counterparts, the iterative Arnoldi and Arnoldi-Chebyshev algorithms. The results show that the refined algorithms are considerably more efficient than their counterparts.
