Abstract
The Lanczos method is often used to solve a large scale symmetric matrix eigenvalue problem. It is well-known that the single-vector Lanczos method can only find one copy of any multiple eigenvalue (unless certain deflating strategy is incorporated) and encounters slow convergence towards clustered eigenvalues. On the other hand, the block Lanczos method can compute all or some of the copies of a multiple eigenvalue and, with a suitable block size, also compute clustered eigenvalues much faster. The existing convergence theory due to Saad for the block Lanczos method, however, does not fully reflect this phenomenon since the theory was established to bound approximation errors in each individual approximate eigenpairs. Here, it is argued that in the presence of an eigenvalue cluster, the entire approximate eigenspace associated with the cluster should be considered as a whole, instead of each individual approximate eigenvectors, and likewise for approximating clusters of eigenvalues. In this paper, we obtain error bounds on approximating eigenspaces and eigenvalue clusters. Our bounds are much sharper than the existing ones and expose true rates of convergence of the block Lanczos method towards eigenvalue clusters. Furthermore, their sharpness is independent of the closeness of eigenvalues within a cluster. Numerical examples are presented to support our claims.
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Notes
If \(k=\ell \), we may say that these angles are between \({\mathcal {X}}\) and \({\mathcal {Y}}\).
In fact, a result due to Chebyshev himself says that if \(p(t)\) is a polynomial of degree no bigger than \(j\) and \(|p(t)|\le 1\) for \(-1\le t\le 1\), then \(|p(t)|\le |{\fancyscript{T}}_j(t)|\) for any \(t\) outside \([-1,1]\) [4, p. 65].
By convention, \(\prod _{j=1}^0(\cdots )\equiv 1\).
A by-product of this is that \(c=1\) if \(k=\ell \).
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Acknowledgments
The authors are grateful to both reviewers for their constructive comments/suggestions that improve the presentation considerably. Li is supported in part by NSF grants DMS-1115834 and DMS-1317330, and a Research Gift grant from Intel Corporation, and and NSFC grant 11428104. Zhang is supported in part by NSFC grants 11101257, 11371102, and the Basic Academic Discipline Program, the 11th five year plan of 211 Project for Shanghai University of Finance and Economics.
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Li, RC., Zhang, LH. Convergence of the block Lanczos method for eigenvalue clusters. Numer. Math. 131, 83–113 (2015). https://doi.org/10.1007/s00211-014-0681-6
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DOI: https://doi.org/10.1007/s00211-014-0681-6