. 2017 Dec 5;17(12):2810. doi: 10.3390/s17122810

Table 4.

Kalman filtering complexity depending on n, m and p.

Algo	Opération	$O (.)$
(E)KF	${\tilde{X}}_{k + 1} = [A] {\hat{X}}_{k} + [B] U_{k}$	$2 n^{2}$
	$[{\tilde{P}}_{k + 1}] = [A] [{\hat{P}}_{k}] {[A]}^{T} + [Q]$	$4 n^{3}$
	${\tilde{Y}}_{k} = [C] {\tilde{X}}_{k} + [D] U_{k}$	$2 m (n + p)$
	$E_{k} = Y_{k} - {\tilde{Y}}_{k}$	m
	$[K_{k + 1}] = [{\tilde{P}}_{k}] {[C]}^{T} {([C] [{\tilde{P}}_{k}] {[C]}^{T} + [R])}^{- 1}$	$4 n^{2} m$ / $4 m^{2} n$
	${\hat{X}}_{k} = {\tilde{X}}_{k} + [K_{k}] E_{k}$	$2 m n$
	$[{\hat{P}}_{k}] = (I - [K_{k}] [C]) [{\tilde{P}}_{k}]$	∼ $2 n^{3}$ / $2 m^{2} n$
UKF	$X_{k} = [{\hat{X}}_{k}, {\hat{X}}_{k} \pm \sqrt{(n + λ) [{\hat{P}}_{k}]}]$	$n^{3}$
	${\tilde{X}}_{k + 1}^{(i)} = f (X_{k}^{(i)})$	$2 n O (f (.))$
	${\tilde{X}}_{k + 1} = \sum_{i = 0}^{2 n} ω_{i}^{μ} {\tilde{X}}_{k + 1}^{(i)}$	$4 n^{2}$
	$[{\tilde{P}}_{k + 1}] = \sum_{i = 0}^{2 n} ω_{i}^{c} ({\tilde{X}}_{k + 1}^{(i)} - {\tilde{X}}_{k + 1}) {({\tilde{X}}_{k + 1}^{(i)} - {\tilde{X}}_{k + 1})}^{T} + [Q]$	$6 n^{3}$
	${\tilde{Y}}_{k + 1}^{(i)} = g ({\tilde{X}}_{k + 1}^{(i)}, U_{k + 1}, 0)$	$(2 n + 1) O (g (.))$
	${\tilde{Y}}_{k + 1} = \sum_{i = 0}^{2 n} ω_{i}^{μ} {\tilde{Y}}_{k + 1}^{(i)}$	$4 m^{2} n$
	$[{\tilde{P}}_{y y, k + 1}] = \sum_{i = 0}^{2 n} ω_{i}^{c} ({\tilde{Y}}_{k + 1}^{(i)} - {\tilde{Y}}_{k + 1}) {({\tilde{Y}}_{k + 1}^{(i)} - {\tilde{Y}}_{k + 1})}^{T} + [R]$	$6 m^{2} n$
	$[{\tilde{P}}_{x y, k + 1}] = \sum_{i = 0}^{2 n} ω_{i}^{c} ({\tilde{X}}_{k + 1}^{(i)} - {\tilde{X}}_{k + 1}) {({\tilde{Y}}_{k + 1}^{(i)} - {\tilde{Y}}_{k + 1})}^{T}$	$4 n^{2} m$
	${\hat{X}}_{k} = {\tilde{X}}_{k} + [K_{k}] E_{k}$	$2 m n$
	$[{\hat{P}}_{k}] = (I - [K_{k}] [C]) [{\tilde{P}}_{k}]$	∼ $2 n^{3}$ / $2 m^{2} n$