ON A NON-STAGNATION CONDITION FOR GMRES AND APPLICATION TO SADDLE POINT MATRICES ∗
Valeria Simoncini · 2010
Abstract. In [25] a new non-stagnation condition for the convergence of GMRES on indefinite problems was proposed. In this paper we derive an enhanced strategy leading to a more general non-stagnation condition. Moreover, we show that the analysis also provides a good setting to derive asymptotic convergence rate estimates for indefinite problems. The analysis is then explored in the context of saddle point matrices, when these are preconditioned in a way so as to lead to nonsym-metric and indefinite systems. Our results indicate that these matrices may represent an insightful training set towards the understanding of the interaction between indefiniteness and stagnation. 1. Introduction. A real n×nmatrix A is said to be positive definite (or positive real) if x⊤Ax> 0 for any real nonzero vector x of length n, where x ⊤ is the transpose of x. A similar definition holds for negative definite matrices. Large nonnormal real linear systems of the form Ax = b are known to be particularly difficult to solve by iterative Krylov subspace methods whenever A is not definite, that is when the