Geodesic
The complete stagnation of GMRES for \(n \leq 4\)
Electronic transactions on numerical analysis, Tome 39 (2012), pp. 75-101
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 \(n \leq 4\)},
     journal = {Electronic transactions on numerical analysis},
     pages = {75--101},
     year = {2012},
     volume = {39},
     zbl = {1321.65049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2012__39__a20/}
}
TY  - JOUR
AU  - Meurant,  Gérard
TI  - The complete stagnation of GMRES for \(n \leq 4\)
JO  - Electronic transactions on numerical analysis
PY  - 2012
SP  - 75
EP  - 101
VL  - 39
UR  - http://geodesic.mathdoc.fr/item/ETNA_2012__39__a20/
LA  - en
ID  - ETNA_2012__39__a20
ER  - 
%0 Journal Article
%A Meurant,  Gérard
%T The complete stagnation of GMRES for \(n \leq 4\)
%J Electronic transactions on numerical analysis
%D 2012
%P 75-101
%V 39
%U http://geodesic.mathdoc.fr/item/ETNA_2012__39__a20/
%G en
%F ETNA_2012__39__a20
Meurant,  Gérard. The complete stagnation of GMRES for \(n \leq 4\). Electronic transactions on numerical analysis, Tome 39 (2012), pp. 75-101. http://geodesic.mathdoc.fr/item/ETNA_2012__39__a20/