The new elements of the j-th column of \( \hat{D}^{-1} \) are then given by:

$$ \begin{equation} d_{kj}^{-1}(\mathbf{r}^{\mathrm{new}}) = d_{kj}^{-1}(\mathbf{r}^{\mathrm{old}}) - \frac{S_j}{R}\,d_{ki}^{-1}(\mathbf{r}^{\mathrm{old}})\quad \begin{array}{ll} \forall\ \ k\in\{1,\dots,N\}\\j\neq i \end{array} \tag{8} \end{equation} $$