The complex eigenvalue analysis is a widely used technique to investigate the stability of a dynamical system with frictional contact. In the case of brake systems, it is the most frequently employed method to study the propensity of the brake to generate squeal noise. When finite element models are considered, iterative solvers are needed to calculate the complex modes and eigenvalues with good precision. In practice, reduced real bases are often used in order to reduce the computational times. However, great attention should be focused on the errors introduced by the reduction, which is rarely done. In this paper, the performances of some reduced bases are investigated in the case of a simple disc/pads system. Bases composed of real coupled modes and bases provided by Component Mode Synthesis (CMS) techniques are tested. An enrichment of these bases is proposed in order to improve the precision of the results. In particular, new rubbing attachment modes are proposed to adapt free-interface CMS techniques to frictional contact. When real coupled modes are used, it is suggested to complete the basis by the static response to a first-order approximation of the friction forces. Applied to the disc/pads model, the different enrichment options allow a reduction of the errors on frequencies, divergence rates and mode shapes by a factor comprised between 10 and 100 without significantly increasing the computational times.