Eq. (21) is now rewritten using the Gauss-Legendre method resulting in $$ \begin{equation} \int_{-1}^{+1}ds\frac{f(\Delta s+x)-f(x)}{s}=\sum_{i=1}^{N}\omega_i\frac{f(\Delta s_i+x)-f(x)}{s_i}, \tag{22} \end{equation} $$ where \( s_i \) are the mesh points (\( N \) in total) and \( \omega_i \) are the weights.

In the selection of mesh points for a PV integral, it is important to use an even number of points, since an odd number of mesh points always picks \( s_i=0 \) as one of the mesh points. The sum in Eq. (22) will then diverge.