We have $$d^{-1}_{kj}(\mathbf{x^{new}}) = d^{-1}_{kj}(\mathbf{x^{old}}) - \frac{d^{-1}_{ki}(\mathbf{x^{old}})}{R} \sum_{l=1}^{N}[\phi_{l}(\mathbf{r_{i}^{new}}) - \phi_{l}(\mathbf{r_{i}^{old}})] d^{-1}_{lj}(\mathbf{x^{old}}),$$ or

$$ \begin{align} d^{-1}_{kj}(\mathbf{x^{new}}) = d^{-1}_{kj}(\mathbf{x^{old}}) \qquad & - & \frac{d^{-1}_{ki}(\mathbf{x^{old}})}{R} \sum_{l=1}^{N}\phi_{l}(\mathbf{r_{i}^{new}}) d^{-1}_{lj}(\mathbf{x^{old}}) \nonumber\\ & + & \frac{d^{-1}_{ki}(\mathbf{x^{old}})}{R} \sum_{l=1}^{N}\phi_{l}(\mathbf{r_{i}^{old}}) d^{-1}_{lj}(\mathbf{x^{old}})\nonumber\\ = d^{-1}_{kj}(\mathbf{x^{old}}) \qquad & - & \frac{d^{-1}_{ki}(\mathbf{x^{old}})}{R} \sum_{l=1}^{N} d_{il}(\mathbf{x^{new}}) d^{-1}_{lj}(\mathbf{x^{old}}) \nonumber\\ & + & \frac{d^{-1}_{ki}(\mathbf{x^{old}})}{R} \sum_{l=1}^{N} d_{il}(\mathbf{x^{old}}) d^{-1}_{lj}(\mathbf{x^{old}}).\nonumber \end{align} $$