Abstract
In this paper we give necessary and sufficient conditions for the complete or partial stagnation of the GMRES iterative method for solving real linear systems. Our results rely on a paper by Arioli, Pták and Strakoš (1998), characterizing the matrices having a prescribed convergence curve for the residual norms. We show that we have complete stagnation if and only if the matrix A is orthonormally similar to an upper or lower Hessenberg matrix having a particular first row or column or a particular last row or column. Partial stagnation is characterized by a particular pattern of the matrix Q in the QR factorization of the upper Hessenberg matrix generated by the Arnoldi process.
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