A complete system of equations sufficient to handle the fully general relativistic hydrodynamics in the case of spherical symmetry is presented, as well as its numerical implementation in a pseudo-spectral code. This code is able to follow tiny radial perturbations of a neutron star near the maximum mass, and especially to show the growth of their fundamental mode as the central density increases, as well as to describe a complete collapse towards a black hole when the instability threshold is passed. In other respects, the code is employed to address the problem of the minimum mass of a black hole formed via gravitational collapse.