Complete stagnation of GMRES for normal matrices

摘要

In this paper we study the problem of complete stagnation of the generalized minimum residual (GMRES) method for normal matrices. We first characterize all n×n nonsingular normal matrices A such that GMRES(A,b) stagnates completely for some vector b. Also we give necessary and sufficient conditions for the non-existence of a real stagnation vector for real normal matrices. The number of real stagnation vectors for normal matrices is studied. Moreover, we characterize all the eigenvalues of nonsingular normal matrices A∈M3(C) such that GMRES(A,b) stagnates completely for some b∈C3. Using the results derived by A. Greenbaum, V. Pták and Z. Strakoš in 1996, we consider the complete stagnation of unitary matrices and derive another characterization for all nonsingular normal matrices A∈M3(R) such that GMRES(A,b) stagnates completely for some vector b∈R3.