## 10 Numerical Experiments
Hence, a potential source of ill-conditioning of the singular subspaces is the closeness between the singular values of the matrix. Another cause of ill-conditioning may be the presence of small singular values which, according to (38), leads to large elements of the vector \(x_{2}\) and large norms of \(\delta U_{1},\ \delta V\). The separation of the parameter vector \(x\) in two parts \(x_{1}\) and \(x_{2}\) reveals the independent importance of these two causes._  
**Example 3**. : _In this example, we compare the perturbation bounds of the singular subspaces derived in this paper with some known bounds from the literature._  
_For the matrix \(A\) and the perturbation \(E\) given in the previous example, compare first the sensitivity of the singular vector \(v_{3}\) associated with the minimum singular value of \(A\) with the respective estimate of \(\sin\Theta_{3}(v_{3},\widehat{v_{3}})\) presented in [Example 2, p. 267] [5]._  
_In Figure 7, we show the exact value of \(|\sin\Theta_{3}|\), the estimate given in [5], and the linear estimate \(\sin\Theta^{lin}\) derived in this paper as functions of the parameter \(\tau\). (The estimate from [5] is valid for \(\tau>\|\delta A\|_{2}\).) Both estimates are very close for all values of \(\tau\).__Compare now the sensitivity of the left \(\mathcal{U}_{2}\) and right \(\mathcal{V}_{2}\) singular subspaces for the same matrix, with the bounds derived in [13] and the nonlinear estimates obtained in this paper. Since the estimates in [13] require knowing the norms of parts of the exact perturbation \(\delta A\), these norms are substituted by \(\|\delta A\|_{2}\) for a fair comparison._  
_In Figure 8, we show the exact value of \(\|\tan\Phi\|_{F}=\sqrt{\tan\Phi_{1}^{2}+\tan\Phi_{2}^{2}}\), the respective estimate from [13], and the estimate \(\|\tan\Phi^{\text{nonl}}\|_{F}=\sqrt{\tan\Phi_{1}^{\text{nonl}^{2}}+\tan\Phi_ {2}^{\text{nonl}^{2}}}\), where \(\Phi_{1}\) and \(\Phi_{2}\) are the canonical angles between \(\mathcal{U}_{2}\) and \(\tilde{\mathcal{U}}_{2}\). The corresponding comparison of the exact value of \(\|\tan\Theta\|_{F}=|\tan\Theta_{1}|\) with the estimate from [13] and the estimate \(\|\tan\Theta^{\text{nonl}}\|_{F}=\tan\Theta_{1}^{\text{nonl}}\) is given in Figure 9. It is seen from the figures that for values of \(\tau\) between \(9\times 10^{-9}\) and \(10^{-6}\) the estimates from [13] produce slightly better results, but for values of \(\tau\) less than \(9\times 10^{-9}\), these estimates exceed significantly the nonlinear estimates obtained in this paper. (Note that the estimates derived in [13] are valid for \(\tau\geq 6.498\times 10^{-9}\).) The comparison of the estimates shows that in case of ill conditioned problems (small values of \(\tau\)) the bound presented in this paper is less conservative than the estimate given in [13]._  
Figure 8: Exact value of \(\|\tan\Phi\|_{F}\) and its perturbation bounds as functions of \(\tau\). Figure 7: Exact value of \(|\sin\Theta_{3}|\) and its perturbation bounds as functions of \(\tau\). **Example 4**. : _This example illustrates the properties of the linear and nonlinear perturbation bounds obtained in the paper._  
_Consider a \(100\times 80\) matrix, taken as_  
\[A=U_{0}\bigg{[} \begin{array}{c}\Sigma_{0}\\ 0\end{array} \bigg{]}V_{0}^{T},\]  
_where \(\Sigma_{0}=\mathrm{diag}(1,1,\ldots,1)\), the matrices \(U_{0}\) and \(V_{0}\) are constructed as proposed in [34],_  
\[U_{0} = M_{2}S_{U}M_{1},\] \[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\] \[e = [1,1,1,\ldots,1]^{T},\;\;f=[1,-1,1,\ldots,(-1)^{m-1}]^{T},\] \[S_{U} = \mathrm{diag}(1,\sigma,\sigma^{2},\ldots,\sigma^{m-1}),\]  
\[V_{0} = N_{2}S_{V}N_{1},\] \[N_{1} = I_{n}-2 gg^{T}/n,\;\;N_{2}=I_{n}-2 hh^{T}/n,\] \[g = [1,1,1,\ldots,1]^{T},\;\;h=[1,-1,1,\ldots,(-1)^{n-1}]^{T},\] \[S_{V} = \mathrm{diag}(1,\tau,\tau^{2},\ldots,\tau^{n-1}),\]  
_and the matrices \(M_{1},M_{2},N_{1},N_{2}\) are Householder reflections.