## 9 Global Perturbation Bounds
### Global Perturbation Bounds of \(\delta U_{1}\) and \(\delta V\)  
Having nonlinear bounds of \(x\), \(y\), \(|\delta W_{U}|\), and \(|\delta W_{V}|\), we may find nonlinear bounds on the perturbations of the entries of \(U_{1}\) and \(V\) according to the relationships  
\[\delta U_{1}^{nonl} = |U|\delta W_{U}^{nonl}, \tag{67}\] \[\delta V^{nonl} = |V|\delta W_{V}^{nonl}. \tag{68}\]  
For the perturbations of the orthogonal matrices of Example 1 in the case of \(\delta A=10^{-5}A_{0}\), we obtain the nonlinear componentwise bounds  
\[\delta U_{1}^{nonl}=10^{-2}\times\left[\begin{array}{cccc}0.0018797&0.0053307 &0.0221710&0.9714224\\ 0.0012678&0.0045195&0.0227162&1.4182521\\ 0.0014492&0.0046330&0.0227031&0.5447168\\ 0.0017643&0.0042429&0.0164150&0.5441097\\ 0.0015182&0.0034345&0.0120243&0.6452928\\ 0.0017076&0.0051060&0.0184868&0.9697071\\ \end{array}\right]\]  
and  
\[\delta V^{nonl}=10^{-3}\times\left[\begin{array}{cccc}0.0110869&0.0341506&0. 1618907&0.0371700\\ 0.0130756&0.0300673&0.0179387&0.1611512\\ 0.0161396&0.0250967&0.0804756&0.1022096\\ 0.0056705&0.0191966&0.0673203&0.0164507\\ \end{array}\right].\]  
These bounds are close to the obtained respective linear estimates \(\delta U_{1}^{lin}\) and \(\delta V^{lin}\) in Section 7.  
Based on (47) and (48), global estimates of the maximum angles between the unperturbed and perturbed singular subspaces of dimension \(r\) can be obtained using the nonlinear bounds \(\delta W_{U}^{nonl}\) and \(\delta W_{V}^{nonl}\) of the matrices \(\delta W_{U}\) and \(\delta W_{V}\), respectively. For the pair of left and right singular subspaces of dimension \(r\), we obtain that  
\[\Phi\max^{nonl}(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}) = \arcsin(\|\delta W_{U+1:m,1r}^{nonl}\|_{2}), \tag{69}\] \[r\leq\min\{m-1,n\},\] \[\Theta\max^{nonl}(\mathcal{V}_{r},\widetilde{\mathcal{V}}_{r}) = \arcsin(\|\delta W_{V+1:n,1:r}^{nonl}\|_{2}),\] (70) \[r\leq n-1.\]  
In Table 8, we give the exact angles between the unperturbed and perturbed left and right singular subspaces of different dimensions and their nonlinear bounds computed using (69) and (70) for the matrix \(A\) from Example 1 and two perturbations \(\delta A=10^{c}\times A_{0},\ c=-10,-5\). The comparison with the corresponding linear bounds in Table 4 shows that the two types of bounds produce close results. As in estimating the other elements of the singular value decomposition, the global perturbation bounds are slightly larger than the corresponding asymptotic estimates but give guaranteed bounds on the changes of the respective elements, although for a limited size of \(\|\delta A\|\).  
\begin{table}
\begin{tabular}{c c c c} \hline \(\boldsymbol{\epsilon}\) & \(\delta\rho_{i}=\|\delta A\|_{2}\) & \(|\delta\sigma_{i}|\) & \(\delta\sigma_{i}^{nonl}\) \\ \hline \(-10\) & \(2.0819812\times 10^{-9}\) & \(\delta\sigma_{1}=0.1543977\times 10^{-8}\) & \(\delta\sigma_{1}=0.1543978\times 10^{-8}\) \\  & & \(\delta\sigma_{2}=0.4299672\times 10^{-10}\) & \(\delta\sigma_{2}=0.4299589\times 10^{-10}\) \\  & & \(\delta\sigma_{3}=0.6114032\times 10^{-9}\) & \(\delta\sigma_{3}=0.6114031\times 10^{-9}\) \\  & & \(\delta\sigma_{4}=0.1736317\times 10^{-9}\) & \(\delta\sigma_{4}=0.1736329\times 10^{-9}\) \\ \hline \(-5\) & \(2.0819812\times 10^{-4}\) & \(\delta\sigma_{1}=0.1543975\times 10^{-3}\) & \(\delta\sigma_{1}=0.1544264\times 10^{-3}\) \\  & & \(\delta\sigma_{2}=0.4299891\times 10^{-5}\) & \(\delta\sigma_{2}=0.4373954\times 10^{-5}\) \\  & & \(\delta\sigma_{3}=0.6113327\times 10^{-4}\) & \(\delta\sigma_{3}=0.6143758\times 10^{-4}\) \\  & & \(\delta\sigma_{4}=0.1737508\times 10^{-4}\) & \(\delta\sigma_{4}=0.2848085\times 10^{-4}\) \\ \hline \end{tabular}
\end{table}
Table 7: Perturbations of the singular values and their nonlinear estimates.