## The complete stagnation of GMRES for $n\le 4$

Gérard Meurant

### Abstract

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 $n$.

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### Key words

GMRES, stagnation, linear systems

15A06, 65F10

### Links to the cited ETNA articles

 [47] Vol. 37 (2010), pp. 202-213 Valeria Simoncini: On a non-stagnation condition for GMRES and application to saddle point matrices