If the new position \( \mathbf{r}^{\mathrm{new}} \) is accepted, then the inverse matrix can by suitably updated by an algorithm having a time scaling of \( O(N^2) \). This algorithm goes as follows. First we update all but the i-th column of \( \hat{D}^{-1} \). For each column \( j\neq i \), we first calculate the quantity:

$$ \begin{equation*} S_j = (\hat{D}(\mathbf{r}^{\mathrm{new}})\times \hat{D}^{-1}(\mathbf{r}^{\mathrm{old}}))_{ij} = \sum_{l=1}^N d_{il}(\mathbf{r}^{\mathrm{new}})\, d^{-1}_{lj}(\mathbf{r}^{\mathrm{old}}) \tag{5} \end{equation*} $$