## 10 Numerical Experiments
The condition numbers of \(U_{0}\) and \(V_{0}\) with respect to the inversion are controlled by the variables \(\sigma\) and \(\tau\) and are equal to \(\sigma^{m-1}\) and \(\tau^{n-1}\), respectively. In the given case, \(\sigma=1.05\), \(\tau=1.1\) and \(\mathrm{cond}(U_{0})=125.2\), \(\mathrm{cond}(V_{0})=186.2\). The minimum singular value of the matrix \(A\) is \(\sigma_{\min}(A)=0.103068025192609\). The perturbation of \(A\) is taken as \(\delta A=10^{c}\times A_{0}\), where \(c\) is a negative number and \(A_{0}\) is a matrix with random entries generated by the MATLAB(r)functionrand._  
_In Figures 10-15, we show several results related to the perturbations of the singular value decomposition of \(A\) for 30 values of \(c\) between \(-12\) and \(-5\). As particular examples, in Figure 10 we display the perturbations of the entry \(U_{60,20}\), which is an element of the matrix \(U_{1}\), and in Figure 11, we display the perturbations of the entry \(V_{60,20}\), both as functions of \(\|\delta A\|_{2}\). The componentwise linear bounds correctly reflect the behavior of the actual perturbations and are valid for wide changes in the perturbation size. Note that this holds for all elements of \(U\) and \(V\). The global (nonlinear) bounds practically coincide with the linear bounds but do not exist for perturbations whose size is larger than \(1.5\times 10^{-4}\)._  
Figure 9: Exact value of \(\|\tan\Theta\|_{F}\) and its perturbation bounds as functions of \(\tau\). Figure 11: Exact values of \(|\delta V_{60,20}|\) and the corresponding linear and nonlinear estimates as functions of the perturbation norm. \[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\]0 Exact values of \(\Phi\max_{90}\) and the corresponding linear and nonlinear estimates as functions of the perturbation norm. \[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\]1 Exact values of \(|\delta U_{60,20}|\) and the corresponding linear and nonlinear estimates as functions of the perturbation norm. \[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\]2 Perturbations of the singular values and their nonlinear bounds for \(\delta A=4.1517085\times 10^{-10}\). \[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\]3 Exact values of \(\Theta\max_{30}\) and the corresponding linear and nonlinear estimates as functions of the perturbation norm. \[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\]4 Perturbations of the singular values and their nonlinear bounds for \(\delta A=2.7913420\times 10^{-5}\). In Figures 12 and 13, we show the angles between the perturbed and unperturbed left_  
\[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\]5  
_and right_  
\[M_{1} = I_{m}-2 ee^{T}/m,\;\;M_{2}=I_{m}-2 ff^{T}/m,\]6  
_singular subspaces of dimension \(50\). Again, the linear bounds on the angles \(|\Phi\max(\mathcal{U}_{50},\widetilde{\mathcal{U}}_{50})|\) and \(|\Theta\max(\mathcal{V}_{50},\widetilde{\mathcal{V}}_{50})|\) are valid for perturbation magnitudes from \(\|\delta A\|_{2}=10^{-10}\) to \(\|\delta A\|_{2}=10^{-4}\) and this also holds for singular subspaces of other dimensions. Note that for sufficiently large \(\|\delta A\|_{2}\), the linear estimate also become non-valid._  
In Figures 14 and 15, we show the perturbations of the singular values and their nonlinear bounds for two different perturbations with \(\|\delta A\|_{2}=2.7282784\times 10^{-10}\) and \(\|\delta A\|_{2}=4.3932430\times 10^{-4}\). While in the first case, the nonlinear bound \(\delta\sigma_{i}^{\text{nonI}}\) is close to the actual change \(\delta\sigma_{i}\) of the singular values, in the second case, the bound becomes significantly greater than the actual change due to the overestimating of the higher-order term \(\Delta_{1}\). From the results shown in Figures 10-13, it follows that for the \(100\times 80\) matrix in the example under consideration, the overestimates of the perturbations of \(U\) and \(V\) are approximately 200 times, while the overestimates of \(\Phi\max\) and \(\Theta\max\) are approximately 100 and 170 times, respectively.