## On a non-stagnation condition for GMRES and application to saddle point matrices

Valeria Simoncini

### Abstract

In Simoncini and Szyld [Numer. Math., 109 (2008), pp. 477–487] 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 nonsymmetric 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.

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### Key words

saddle point matrices, large linear systems, GMRES, stagnation.

### AMS subject classifications

65F10, 65N22, 65F50.

### Links to the cited ETNA articles

 [15] Vol. 12 (2001), pp. 205-215 Bernd Fischer and Franz Peherstorfer: Chebyshev approximation via polynomial mappings and the convergence behaviour of Krylov subspace methods [21] Vol. 22 (2006), pp. 114-121 Chen Greif and Dominik Schötzau: Preconditioners for saddle point linear systems with highly singular (1,1) blocks

### ETNA articles which cite this article

 Vol. 39 (2012), pp. 75-101 Gérard Meurant: The complete stagnation of GMRES for $n\le 4$ Vol. 40 (2013), pp. 381-406 Desire Nuentsa Wakam and Jocelyne Erhel: Parallelism and robustness in GMRES with a Newton basis and deflated restarting

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