of the $\infty$-replica symmetry breaking equations. We have developed a new numerical approach based on a pseudo-spectral code which leads to a strong enhancement of the quality of the numerical results. We have also shown how, for example, to determine the value of $x_{\rm max}$ numerically. In all cases the agreement between the numerical and the analytical results is rather good.

We conclude by stressing that our results go beyond the interest on the Sherrington - Kirkpatrick model, since the method we used here are far more general and can be employed to a wider class of models with generalized $\infty$-replica symmetry breaking equations such as those introduced in Ref. [@Luca]. In particular in this reference the numerical method was applied to the 3-SAT model, and the extension to other relevant models is under development.

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