# Table 7 Complexity of algorithm II

Stage Computational unit Computational complexity of each unit Total computational complexity
Stage 1 The same as algorithm I The same as algorithm I $${\displaystyle \begin{array}{c}{K}_1\left(\begin{array}{c}3D\left(M-1\right)\left(3{M}^2+M\right)\\ {}+3{D}^2M{\left(M-1\right)}^2+18{M}^2+O\left({(3M)}^3\right)\end{array}\right)\\ {}+{K}_2\left(\begin{array}{c}3M\left(M-1\right)\left(3M+N-1\right)\\ {}+3\left(M+3\right)\left(M+N-2\right)\\ {}+\left(N+3\right){\left(M+N-2\right)}^2\\ {}+3M{\left(M-1\right)}^2+O\left({\left(N-1\right)}^3\right)\\ {}+2\cdot O\left({\left(M+N-2\right)}^3\right)+O(27)\end{array}\right)\\ {}+2{D}^2\left(N-1\right){\left(M-1\right)}^2+3M{\left(M-1\right)}^2\\ {}+\left(N-1\right)\left(N+M-2\right)+2{\left(N-1\right)}^2\\ {}+\left(N-1\right)\left(M-1\right)\left(2N+M-3\right)\\ {}+3 DM\left(M-1\right)\left(3M+N-1\right)\\ {}+3M\left(M-1\right)\left(3M+N-1\right)\\ {}+D\left(2N-1\right)\left(N-1\right)\left(M-1\right)\\ {}+O\left({(3M)}^3\right)+2\cdot O\left({\left(N-1\right)}^3\right)\\ {}+2\cdot O\left({\left(D\left(M-1\right)\right)}^3\right)\end{array}}$$(where K1 and K2 denote the iteration numbers for the dimension-reduction Taylor-series iterative algorithms in the first and second stages, respectively)
Stage 2 Φ 3DM(M − 1)(3M + N − 1)
$${\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1}\right\}}_{1\le k\le {K}_2}$$ and $${\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\right\}}_{1\le k\le {K}_2}$$ $${K}_2\left(\begin{array}{c}3M\left(M-1\right)\left(3M+N-1\right)\\ {}+3M{\left(M-1\right)}^2+2\cdot O\left({\left(M+N-2\right)}^3\right)\end{array}\right)$$
$${\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\left[\begin{array}{c}{\mathbf{F}}_1\left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\\ {}{\mathbf{O}}_{\left(N-1\right)\times 3}\end{array}\right]\right\}}_{1\le k\le {K}_2}$$ 3K2(N + M − 2)(M − 1)
$${\left\{{\boldsymbol{\Pi}}^{\perp}\left({\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]\right)\right\}}_{1\le k\le {K}_2}$$ $${K}_2\left(\begin{array}{c}\left(N-1\right){\left(N+M-2\right)}^2\\ {}+O\left({\left(N-1\right)}^3\right)\end{array}\right)$$
$${\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\left[\begin{array}{c}\hat{\mathbf{r}}-\mathbf{f}\left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\\ {}{\hat{\boldsymbol{\uprho}}}_{\mathrm{f}}\end{array}\right]\right\}}_{1\le k\le {K}_2}$$ K2(N + M − 2)2
$${\left\{{\hat{\mathbf{u}}}_{\mathrm{s}2}^{\left(k+1\right)}\right\}}_{1\le k\le {K}_2}$$ $${K}_2\left(\begin{array}{c}12\left(N+M-2\right)\\ {}+3{\left(N+M-2\right)}^2+O(27)\end{array}\right)$$
$${\left({\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]}^{\mathrm{T}}{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1}\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]\right)}^{-1}$$ $${\displaystyle \begin{array}{c}3M\left(M-1\right)\left(3M+N-1\right)\\ {}+\left(N-1\right)\left(M-1\right)\left(2N+M-3\right)\\ {}+3M{\left(M-1\right)}^2+O\left({\left(N-1\right)}^3\right)\end{array}}$$
$${\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]}^{\mathrm{T}}{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1}\left[\begin{array}{c}\hat{\mathbf{r}}-\mathbf{f}\left({\hat{\mathbf{u}}}_{\mathrm{s}2},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\\ {}{\hat{\boldsymbol{\uprho}}}_{\mathrm{f}}\end{array}\right]$$ (N − 1)(N + M − 2)
$${\hat{\boldsymbol{\uprho}}}_{\mathrm{s}2}$$ (N − 1)2