Krylov subspace methods in the electronic industry

In most practical cases, the convergence of the GMRES method applied to a linear algebraic system Ax -- b is determined by the distribution of eigenvalues of A. 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 matrix A has the prescribed eigenvalues. Moreover, a characterization of the right hand sides b for which the GMRES(A, b) converges exactly in m steps, where m is the degree of the minimal polynomial of A, is given