  In this paper, we investigate the Anderson model on the Bethe lattice,
focusing on the localized regime. Employing the cavity approach, we derive
compact expressions for the inverse participation ratios (IPRs) that are
equivalent to those obtained using the supersymmetric formalism and naturally
facilitate a highly efficient computational scheme. This method yields
numerical results with unprecedented accuracy, even very close to the
localization threshold. Our approach allows for high-precision validation of
all theoretical predictions from the analytical solution, including the finite
jump of the IPRs at the transition. Additionally, we reveal a singular behavior
of the IPRs near the critical point that has not been previously reported in
the literature. This singular behavior is further confirmed by the numerical
solution of the non-linear $\sigma$ model on the Bethe lattice, which provides
an effective description of Anderson localization.
