Algorithm 1 SVD based.

1 Arrange the data of snapshot matrix S into two different matrices, as: $$S^{\prime }\triangleq \left[y^1\dots y^{m-1}\right];S^{″}\triangleq \left[y^2\dots y^m\right]$$ 2 Compute the SVD of $S^{\prime }$; $S^{\prime }=U\Sigma V^{*}$ where, $U=\left[u_1,u_2,\dots ,u_r\right]\in {ℝ}^{n\times r}$ and $V=\left[v_1,v_2,\dots ,v_r\right]\in {ℝ}^{m\times r}$ are orthogonal matrices and $\Sigma =\mathrm{diag}\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r\right)\in {ℝ}^{r\times r}$ is a diagonal matrix. 3 Define a matrix $\widetilde{A_y}$ such that, $\widetilde{A_y}\triangleq U^{*}{\Sigma }^{-1}V{S}^{″}$; here, $\widetilde{A_y}$ is the $k\times k$ projection of $A_y$ such that, $\widetilde{A_y}$ = $U^{*}{A}_{y}U$. 4 Compute the eigenvalues and eigenvectors of $\widetilde{A_y}$ as: $\widetilde{A_y}w=\lambda w$. 5 Calculate the DMD mode corresponding to λ: ${\mathrm{\Psi }}^{\mathrm{DMD}}=S^{″}V{\Sigma }^{-1}w$.