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Generalization of convergence conditions for a restarted GMRES
Author(s) -
Zítko Jan
Publication year - 2000
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/(sici)1099-1506(200004/05)7:3<117::aid-nla189>3.0.co;2-z
Subject(s) - generalized minimal residual method , mathematics , hermitian matrix , convergence (economics) , generalization , matrix (chemical analysis) , linear system , coefficient matrix , pure mathematics , mathematical analysis , eigenvalues and eigenvectors , materials science , physics , quantum mechanics , economics , composite material , economic growth
We consider the GMRES( s ), i.e. the restarted GMRES with restart s for the solution of linear systems Ax = b with complex coefficient matrices. It is well known that the GMRES( s ) applied on a real system is convergent if the symmetric part of the matrix A is positive definite. This paper introduces sufficient conditions implying the convergence of a restarted GMRES for a more general class of non‐Hermitian matrices. For real systems these conditions generalize the known result initiated as above. The discussion after the main theorem concentrates on the question of how to find an integer j such that the GMRES( s ) converges for all s ≥ j . Additional properties of GMRES obtained by derivation of the main theorem are presented in the last section. Copyright © 2000 John Wiley & Sons, Ltd.