Determining a determinant of an $N \times N$ matrix by standard Gaussian elimination is of the order of $\mathbf{O}(N^3)$ calculations. As there are $N\cdot d$ independent coordinates we need to evaluate $Nd$ Slater determinants for the gradient (quantum force) and $N\cdot d$ for the Laplacian (kinetic energy)

With the updating algorithm we need only to invert the Slater determinant matrix once. This is done by calling standard LU decomposition methods.