We consider the approximation of the field of values of the inverse of a large sparse matrix, without explicitly computing the inverse or using its action (i.e., accurately solving a linear system with this matrix). We review results by Manteuffel and Starke and give an alternative that may yield better approximations in practice. We give connections with the harmonic Rayleigh-Ritz approach. Several properties and applications of the studied concepts as well as numerical examples are provided.
Key words: Field of values, numerical range, matrix inverse, large sparse matrix, Ritz values, harmonic Rayleigh-Ritz, harmonic Ritz values, GMRES convergence, Arnoldi, numerical radius, numerical abscissa, inner numerical radius, inclusion region.