Convergence of the block Lanczos method for eigenvalue clusters

Ren-Cang Li University of Texas at Arlington Lei-hong Zhang Shanghai University of Finance and Economics

Numerical Linear Algebra mathscidoc:1808.26001

Distinguished Paper Award in 2019

Numerische Mathematik, 131, 83-113, 2015.10
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 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.
Block Lanczos, Eigenvalue, cluster, convergence, eigenspace
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  title={Convergence of the block Lanczos method for eigenvalue clusters},
  author={Ren-Cang Li, and Lei-hong Zhang},
  booktitle={Numerische Mathematik},
Ren-Cang Li, and Lei-hong Zhang. Convergence of the block Lanczos method for eigenvalue clusters. 2015. Vol. 131. In Numerische Mathematik. pp.83-113.
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