This equation scales as $O(N^2)$. The evaluation of the determinant of an $N \times N$ matrix by standard Gaussian elimination requires $\mathbf{O}(N^3)$ calculations. As there are $Nd$ independent coordinates we need to evaluate $Nd$ Slater determinants for the gradient (quantum force) and $Nd$ for the Laplacian (kinetic energy). With the updating algorithm we need only to invert the Slater determinant matrix once. This can be done by standard LU decomposition methods.