Predict Step: $k= \{1,2,..., n\} $
$\mathbf{\hat{x}}_{k|k-1}=\mathbf{F}_{k-1}\mathbf{\hat{x}}_{k-1|k-1}+\mathbf{g}_{k-1}$
$\mathbf{\hat{P}}_{k|k-1}=\mathbf{F}_{k-1}\mathbf{\hat{P}}_{k-1|k-1}\mathbf{F}^\mathrm{T}_{k-1}+\mathbf{Q}_{k-1}$
Update Step: $k= \{1,2,..., n\} $
$\mathbf{y}_k=\mathbf{z}_{k}-\mathbf{H}\mathbf{\hat{x}}_{k|k-1}$
$\mathbf{K}_k=\mathbf{{P}}_{k|k-1}\mathbf{H}^\mathrm{T}_{k}(\mathbf{\mathbf{H}_{k}\mathbf{{P}}_{k|k-1}\mathbf{H}^\mathrm{T}_{k}+\mathbf{R}_{k}})^{-1}$
$\mathbf{\hat{x}}_{k|k}=\mathbf{\hat{x}}_{k|k-1}+\mathbf{K}_{k}\mathbf{y}_{k}$
$\mathbf{{P}}_{k|k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{{P}}_{k|k-1}$
RTS Smoother: $k= \{n-1,n-2,..., 1\} $
$\textbf{C}_{k} = \textbf{P}_{k|k}\textbf{F}_{k+1}^{T}\textbf{P}_{k+1|k}^{-1}$
$\hat{\textbf{x}}_{k|n} = \hat{\textbf{x}}_{k|k} + \textbf{C}_{k}(\hat{\textbf{x}}_{k+1|n} - \hat{\textbf{x}}_{k+1|k}) $
$\textbf{P}_{k|n} = \textbf{P}_{k|k} + \textbf{C}_{k}(\textbf{P}_{k+1|n} - \textbf{P}_{k+1|k})\textbf{C}_{k}^{T}$
What is the time complexity of the above algorithms if $\mathbf{x_k} \in R^{n \times 1}$, $ \mathbf{z_k} \in R^{m \times 1} $, $ \mathbf{H_k} \in R^{m \times n}$, $ \mathbf{F_k} \in R^{n \times n}$, $ \mathbf{P_k,Q_k} \in R^{n \times n}$, $ \mathbf{R_k} \in R^{m \times m}$. Please provide the steps on how to calculate complexity for such systems
Thanks