## 10 Numerical Experiments  
In this section, we present the results of some numerical experiments illustrating the properties of the asymptotic and global estimates obtained in the paper. The computations are performed with MATLAB(r)Version 9.9 (R2020b) [29] using IEEE double-precision arithmetic and are verified by using GNU Octave, v. 5.2.0. The M-files implementing the linear and nonlinear SVD perturbation estimates along the example files are available from the authors. **Example 2**. : _This example illustrates the ill-conditioning of the singular subspaces in the case of close singular values of the given matrix._  
_Consider a \(10\times 3\) matrix \(A\) obtained as_  
\[A=U_{0}\Sigma_{0}V_{0}^{\mathrm{T}},\]  
_where_  
\[U_{0} = I-2 ee^{\mathrm{T}}/10,\;\;e=[1,1,\ldots,1],\] \[V_{0} = I-2 hh^{\mathrm{T}}/3,\;\;h=[1,-1,1]\]  
_are orthogonal ad symmetric matrices (elementary reflections [Ch. 4] [2]) and_  
\[\mathrm{diag}(\Sigma_{0})=(2,1+\tau,1),\]  
_where the parameter \(\tau\) varies between \(10^{-8}\) and \(10^{-1}\)._  
_In Figure 5, we present the actual values of \(\|\delta U_{1}\|_{2}\), \(\|x\|_{2}\) and the angle \(|\Theta\max_{2}|\) along with the asymptotic normwise estimate \(\|\delta U_{1}^{lin}\|_{2}\) as functions of the difference \(\tau=\sigma_{2}-\sigma_{3}=10^{-8},\ldots,10^{-1}\). Similarly, in Figure 6, we show the values of \(\|\delta V\|_{2}\), \(\|y\|_{2}\) and the angle \(|\Theta\max_{2}|\) together with the asymptotic estimate \(\|\delta V^{lin}\|_{2}\) as functions of \(\tau\). Note that \(\Phi\max_{2}\) is the larger angle between the subspaces \(\mathcal{U}_{(2)}=\mathrm{span}(u_{1},u_{2})\) and \(\tilde{\mathcal{U}}_{(2)}=\mathrm{span}(\vec{u}_{1},\vec{u}_{2})\), while \(\Theta\max_{2}=\Theta\max(\vec{v}_{2},\tilde{\mathcal{V}}_{2})\)._ \begin{table}
\begin{tabular}{c c c c} \hline \(c\) & \(\|\delta A\|_{2}\) & \(|\Phi\max(\mathcal{U}_{\mathbf{\tau}},\tilde{\mathcal{U}}_{\mathbf{\tau}})|\) & \(\Phi\max^{\text{t}onl}(\mathcal{U}_{\mathbf{\tau}},\tilde{\mathcal{U}}_{\mathbf{\tau}})\) \\ \hline \(-10\) & \(2.0819813\times 10^{-9}\) & \(\Phi_{1}=0.0024212\times 10^{-8}\) & \(\Phi_{1}=0.0002029\times 10^{-6}\) \\  & & \(\Phi_{2}=0.0082136\times 10^{-8}\) & \(\Phi_{2}=0.0005938\times 10^{-6}\) \\  & & \(\Phi_{3}=0.1032901\times 10^{-8}\) & \(\Phi_{3}=0.0029428\times 10^{-6}\) \\  & & \(\Phi_{4}=1.0746434\times 10^{-8}\) & \(\Phi_{4}=0.1662788\times 10^{-6}\) \\ \hline \(-5\) & \(2.0819812\times 10^{-4}\) & \(\Phi_{1}=0.0024212\times 10^{-3}\) & \(\Phi_{1}=0.0020601\times 10^{-2}\) \\  & & \(\Phi_{2}=0.0082136\times 10^{-3}\) & \(\Phi_{2}=0.0060635\times 10^{-2}\) \\  & & \(\Phi_{3}=0.1032951\times 10^{-3}\) & \(\Phi_{3}=0.0303251\times 10^{-2}\) \\  & & \(\Phi_{4}=1.0734956\times 10^{-3}\) & \(\Phi_{4}=1.6837810\times 10^{-2}\) \\ \hline \(c\) & \(\|\delta A\|_{2}\) & \(|\Theta\max(\mathbf{\gamma}_{\mathbf{\tau}},\tilde{\mathcal{V}}_{\mathbf{\tau}})|\) & \(\Theta\max^{\text{t}onl}(\mathcal{V}_{\mathbf{\tau}},\tilde{\mathcal{V}}_{\mathbf{ \tau}})\) \\ \hline \(-10\) & \(2.0819813\times 10^{-9}\) & \(\Theta_{1}=0.0425801\times 10^{-9}\) & \(\Theta_{1}=0.0166760\times 10^{-8}\) \\  & & \(\Theta_{2}=0.0438355\times 10^{-9}\) & \(\Theta_{2}=0.0438688\times 10^{-8}\) \\  & & \(\Theta_{3}=0.2782232\times 10^{-9}\) & \(\Theta_{3}=0.1714819\times 10^{-8}\) \\ \hline \(-5\) & \(2.0819812\times 10^{-4}\) & \(\Theta_{1}=0.0425804\times 10^{-4}\) & \(\Theta_{1}=0.0166819\times 10^{-3}\) \\  & & \(\Theta_{2}=0.0438356\times 10^{-4}\) & \(\Theta_{2}=0.0438972\times 10^{-3}\) \\  & & \(\Theta_{3}=0.2782378\times 10^{-4}\) & \(\Theta_{3}=0.1718211\times 10^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 8: Nonlinear sensitivity estimates of the singular subspaces. _The results shown in Figures 5 and 6 confirm that the sensitivity of SVD increases with the decreasing distance between the singular values. According to (33), (34), (36) and (37), with the decreasing of \(\tau\) the conditioning of the singular subspaces worsens and the norms of the perturbations \(\delta U_{1},\ \delta V\) increase.