## 4 Equations for the Perturbation Parameters
This observation aligns with the well known fact that the sensitivity of a singular subspace is inversely proportional to the smallest singular value associated with that subspace. As a result of determining the linear estimates (36)-(38), we obtain an asymptotic approximation of the vector \(x\) as  
\[x^{lin}=\Omega_{1}^{\mathrm{T}}x_{(1)}^{lin}+\Omega_{2}^{\mathrm{T}}x_{(2)}^{ lin}. \tag{39}\]  
It should be emphasized that the determination of the linear bounds \(x^{lin}\) and \(y^{lin}\) requires knowing only the norm of the perturbation \(\delta A\). Thanks to the diagonal structure of the matrices \(S_{1},\ S_{2}\), and \(\Sigma\), the solutions to the Equations (26)-(30) are computed efficiently with high accuracy. In fact, the computing of the diagonal elements of the matrices \(S_{xf},S_{xg},S_{yf},S_{yg}\) requires \(10q\) floating point operations (flops), the determining of \(x_{(1)}^{lin}\) and \(y^{lin}\) according to (36) and (37) requires \(4q\) flops, and the obtaining of \(x_{(2)}^{lin}\) by (38) needs \(2(m-n)n\) flops. Thus, the computing of \(x^{lin}\) and \(y^{lin}\) requires \(14n(n-1)/2+2(m-n)n\) flops. Also, the solution of the diagonal systems of equations is performed very accurately in a floating point arithmetic. Consider the \(6\times 4\) matrix  
\[A=\left[ \begin{array}{cccc}3&3&-3&-6\\ -3&-1&1&8\\ 3&1&0&-9\\ -3&1&-2&11.1\\ 6&4&-6&-11.9\\ 3&1&1&-10.1\end{array} \right]\]and assume that it is perturbed by_  
\[\delta A = 10^{c}\times A_{0},\] \[A_{0} = \left[ \begin{array}{rrrr}-5&3&1&-2\\ 7&-4&-9&-5\\ -3&8&5&4\\ 2&-6&-3&-7\\ -5&3&1&9\\ -4&-8&-2&6\end{array} \right],\]  
_where \(c\) is a varying parameter. (For convenience, the entries of the matrix \(A_{0}\) are taken as integers)._  
_The singular value decompositions of the matrices \(A\) and \(A+\delta A\) are computed by the function_ svd _of MATLAB(r)[29]__. The singular values of \(A\) are_  
\[\sigma_{1} = 25.460186918120350,\] \[\sigma_{2} = 7.684342946021752,\] \[\sigma_{3} = 1.248776186923002,\] \[\sigma_{4} = 0.017709242587950.\]  
_In the given case, the matrices \(S_{1}\) and \(S_{2}\) in Equations (27) and (28) are_  
\[S_{1} = \mathrm{diag}(25.4601869,\;25.4601869,\;25.4601869,\;25.4601869,\] \[\qquad 7.6843429,\;7.6843429,\;1.2487762),\] \[S_{2} = \mathrm{diag}(7.6843429,\;1.2487762,\;0.0177092,\] \[\qquad 1.2487762,\;0.0177092,\;0.0177092),\]  
_and the matrices participating in (31) and (32) are_  
\[S_{xf} = \mathrm{diag}(-0.0432135,\;-0.0393717,\;-0.0392770,\] \[\qquad-0.1336647,\;-0.1301354,\;-0.8009451),\] \[\sigma_{2} = 7.684342946021752,\]0 \[\sigma_{2} = 7.684342946021752,\]1 \[\sigma_{2} = 7.684342946021752,\]2 \[\sigma_{2} = 7.684342946021752,\]3  
_The matrix_  
\[\left[ \begin{array}{rr}-S_{1}&S_{2}\\ S_{2}&-S_{1}\end{array} \right],\]  
_which determines the solution for \(x_{(1)}\) and \(y\), has a condition number with respect to the 2-norm equal to 26.9234179. The exact parameters \(x_{k}\) and their linear approximations \(x_{k}^{lin}\) computed by using (36) and (38) are shown to eight decimal digits for two perturbation sizes in Table 1. The differences between the values of \(x_{k}^{lin}\) and \(x_{k}\) are due to the bounding of the elements of the vectors \(f\) and \(g\) by the value of \(\|\delta A\|_{2}\) and taking the terms in (36)-(38) with positive signs. Both approximations are necessary to ensure that \(x\preceq x^{lin}\) for arbitrary small size perturbation._