Premium
Convergence estimates of nonrestarted and restarted block‐Lanczos methods
Author(s) -
Zhou Ming
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2182
Subject(s) - lanczos resampling , mathematics , eigenvalues and eigenvectors , krylov subspace , lanczos algorithm , block (permutation group theory) , generalized minimal residual method , convergence (economics) , subspace topology , matrix (chemical analysis) , invariant subspace , hessian matrix , iterative method , algorithm , mathematical optimization , combinatorics , mathematical analysis , linear subspace , pure mathematics , physics , materials science , quantum mechanics , economics , composite material , economic growth
Summary The block‐Lanczos method serves to compute a moderate number of eigenvalues and the corresponding invariant subspace of a symmetric matrix. In this paper, the convergence behavior of nonrestarted and restarted versions of the block‐Lanczos method is analyzed. For the nonrestarted version, we improve an estimate by Saad by means of a change of the auxiliary vector so that the new estimate is much more accurate in the case of clustered or multiple eigenvalues. For the restarted version, an estimate by Knyazev is generalized by extending our previous results on block steepest descent iterations and single‐vector restarted Krylov subspace iterations. The new estimates can also be reformulated and applied to invert‐block‐Lanczos methods for solving generalized matrix eigenvalue problems.