An Augmented Stability Result for the Lanczos Hermitian Matrix Tridiagonalization Process
SIAM Journal on Matrix Analysis and Applications
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The Lanczos algorithm can be considered as an iterative method for finding a few eigenvalues and eigenvectors of large sparse symmetric matrices. In the present paper we solve a conjecture posed by Zdenek Strakos and Anne Greenbaum in [Open Questions in the Convergence Analysis of the Lanczos Process for the Real Symmetric Eigenvalue Problem, IMA Research Report, 1992] on the clustering of Ritz values, which occurs in finite precision computations. In particular, we prove that the conjecture is valid in most cases and describe the rare case when it is not. The established upper bounds measuring the quality of Ritz approximations imply also that Ritz values cluster only close to an eigenvalue.