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.