Geodesic
The complete stagnation of GMRES for $n \le 4$
Electronic transactions on numerical analysis, Tome 39 (2012), pp. 75-101.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We study the problem of complete stagnation of the generalized minimum residual method for real matrices of order $n\le 4$ when solving nonsymmetric linear systems $Ax=b$. We give necessary and sufficient conditions for the non-existence of a real right-hand side $b$ such that the iterates are $x^k=0,\, k=0,\dots,n-1,$ and $x^n=x$. We illustrate these conditions with numerical experiments. We also give a sufficient condition for the non-existence of complete stagnation for a matrix $A$ of any order $n$.
Classification : 15A06, 65F10
Keywords: GMRES, stagnation, linear systems 
@article{ETNA_2012__39__a20,
     author = {Meurant, G\'erard},
     title = {The complete stagnation of {GMRES} for $n \le 4$},
     journal = {Electronic transactions on numerical analysis},
     pages = {75--101},
     publisher = {mathdoc},
     volume = {39},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2012__39__a20/}
}
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Meurant, Gérard. The complete stagnation of GMRES for $n \le 4$. Electronic transactions on numerical analysis, Tome 39 (2012), pp. 75-101. http://geodesic.mathdoc.fr/item/ETNA_2012__39__a20/