Algorithm 3 Procedure of Computation

1:functionCompute(${\sigma }_x$,$f_{LU}$) 2:    $\{{\mathit{L}}_{\mathbf{1}},{\mathit{U}}_{\mathbf{1}}\}\leftarrow f_{LU}\left({\mathit{Y}}_{\mathbf{1}}\right)$. 3:    $\{{\mathit{L}}_{\mathbf{2}},{\mathit{U}}_{\mathbf{2}}\}\leftarrow f_{LU}\left({\mathit{Y}}_{\mathbf{2}}\right)$. 4:    $\{{\mathit{L}}_{\mathbf{3}},{\mathit{U}}_{\mathbf{3}}\}\leftarrow f_{LU}\left({\mathit{Y}}_{\mathbf{3}}\right)$. 5:    $\{{\mathit{L}}_{\mathbf{4}},{\mathit{U}}_{\mathbf{4}}\}\leftarrow f_{LU}\left({\mathit{Y}}_{\mathbf{4}}\right)$. 6:    $r_1\leftarrow {\prod }_{i=1}^n{L}_1(i,i)U_1(i,i)$. 7:    $r_2\leftarrow {\prod }_{i=1}^n{L}_2(i,i)U_2(i,i)$. 8:    $r_3\leftarrow {\prod }_{i=1}^n{L}_3(i,i)U_3(i,i)$. 9:    $r_4\leftarrow {\prod }_{i=1}^n{L}_4(i,i)U_4(i,i)$. 10:    return ${\sigma }_y=\{r_1,r_2,r_3,r_4,{\mathit{L}}_{\mathbf{1}},{\mathit{L}}_{\mathbf{3}},{\mathit{U}}_{\mathbf{1}},{\mathit{U}}_{\mathbf{3}}\}$ 11: end function