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:

$$\begin{equation} S_j = (\hat{D}(\mathbf{r}^{\mathrm{new}})\times \hat{D}^{-1}(\mathbf{r}^{\mathrm{old}}))_{ij} = \sum_{l=1}^N d_{il}(\mathbf{r}^{\mathrm{new}})\, d^{-1}_{lj}(\mathbf{r}^{\mathrm{old}}) \tag{7} \end{equation}$$