Theoretical error bounds on the convergence of the Lanczos and block-Lanczos methods

In this paper, the new theoretical error bounds on the convergence of the Lanczos and the block-Lanczos methods are established based on results given by Saad [1]. Similar further inequalities are found for the eigenelements by using bounds on the acute angle between the exact eigenvectors and the Krylov subspace spanned by x0, Ax0, …, An−1x0, where x0 is the initial starting vector of the process. The same analysis is extended to the block-Lanczos method. Several numerical experiments are presented in order to permit a comparison between the actual rates of convergence of the Lanczos method with the theoretical error bounds. © 1999 Elsevier Science Ltd. All rights reserved.