Krylov sequences of maximal length and convergence of GMRES

In most practical cases, the convergence of the GMRES method applied to a linear algebraic systemAx=b is determined by the distribution of eigenvalues ofA. In theory, however, the information about the eigenvalues alone is not sufficient for determining the convergence. In this paper the previous work of Greenbaum et al. is extended in the following direction. It is given a complete parametrization of the set of all pairs {A, b} for which GMRES(A, b) generates the prescribed convergence curve while the matrixA has the prescribed eigenvalues. Moreover, a characterization of the right hand sidesb for which the GMRES(A, b) converges exactly inm steps, wherem is the degree of the minimal polynomial ofA, is given.