the previous described procedure of non-commutative geometry to compute the spectral action. We notice that matrix algebra introduces an extension of the standard potential in the commutative case, in fact the scalar field $\phi$ and the $X_\mu$ fields are present together in a potential of the form[^1] $(\alpha X_\mu \star X^\mu  + \beta \bar{\varphi} \star \varphi - 1)^2$, with $\alpha, \beta \in \mathbb{R}^+$ and $X_\mu(x) = (\Theta^{-1} )_{\mu}^\nu x_\nu + A_\mu(x)$ is a covariant coordinate.

The high non-triviality of the vacuum makes very difficult to explicit the vacuum configuration of the system in [@Goursac1] A. de Goursac, J.C. Wallet, and R. Wulkenhaar, using the matrix base formalism, have found an expressions from vacuum solutions deriving them from the relevant solutions of the equations of motion. Although, the complexity of the vacuum configuration makes the perturbative approach very complicated, in order to conduct some investigations will be considered a non-perturbative scheme using a discretized matrix model of the action in which the fields become matrices, the star product become the matrix multiplication and the integral turns in a matrix trace.

Now comes in to play the numerical treatment, the standard method is to approximate the space by discrete points, for example 