Premium
Convergence conditions for a restarted GMRES method augmented with eigenspaces
Author(s) -
Zítko Jan
Publication year - 2005
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.421
Subject(s) - generalized minimal residual method , krylov subspace , mathematics , residual , subspace topology , eigenvalues and eigenvectors , convergence (economics) , norm (philosophy) , dimension (graph theory) , invariant subspace , linear system , linear subspace , iterative method , mathematical optimization , algorithm , mathematical analysis , combinatorics , pure mathematics , physics , quantum mechanics , political science , law , economics , economic growth
We consider the GMRES( m , k ) method for the solution of linear systems Ax = b , i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A ‐invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES( m , k ) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES( m , k ) and illustrate that these augmentation techniques can remove stagnation of GMRES( m ) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd.