Finally the elements of the i-th column of $\hat{D}^{-1}$ are updated simply as follows:

$$\begin{equation} d_{ki}^{-1}(\mathbf{r}^{\mathrm{new}}) = \frac{1}{R}\,d_{ki}^{-1}(\mathbf{r}^{\mathrm{old}})\quad \forall\ \ k\in\{1,\dots,N\} \tag{9} \end{equation}$$

We see from these formulas that the time scaling of an update of $\hat{D}^{-1}$ after changing one row of $\hat{D}$ is $O(N^2)$.

The scheme is also applicable for the calculation of the ratios involving derivatives. It turns out that differentiating the Slater determinant with respect to the coordinates of a single particle $\mathbf{r}_i$ changes only the i-th row of the corresponding Slater matrix.