Generalization of convergence conditions for a restarted GMRES
Jan Z�tko · Numerical Linear Algebra with Applications · 2000
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.