Previous numerical studies of the $\infty$-replica symmetry breaking solution used a naive integration scheme based on the direct discretization of the Parisi’s equation [@VTP81; @SDJPC84; @topomoto; @B90]. The main disadvantages of this approach are the large amount of memory needed for a good resolution of the solution and the numerical problems arising when $\dot{q}(x)$ is small. To overcome these problems we developed a new numerical scheme based on a pseudo-spectral algorithm which allows for rather accurate results for all temperatures with a reasonable amount of memory. Moreover the use of pseudo-spectral methods makes the whole code rather fast.

We stress that while the methods we are going to discuss are applied here to the Sherrington - Kirkpatrick model, they have a wider range of application. In principle can be applied to any model with $\infty$-replica symmetry breaking type solution [@Luca].

We find that for the Sherrington - Kirkpatrick model the Parisi solution $q(x)$ is not an odd function as one may expect from its physical meaning. At any $T < T_c$, the Taylor expansion of $q(x)$ in powers of $x$ contains both odd as well as even powers of $x$. The only term which is missing is $x^2$. The presence 