## 9 Global Perturbation Bounds
### Iterative Procedure for Finding Global Bounds of \(x\) and \(y\)
In the numerical experiments presented below, the initial conditions are chosen as \(x_{0}^{nonl}=eps[1,1,\ldots,1]^{T}\) and \(y_{0}^{nonl}=eps[1,1,\ldots,1]^{T}\), where \(eps\) is the MATLAB(r) function eps, \(eps=2^{-52}\). The stopping criteria for \(x\)- and \(y\)-iterations are taken as  
\[err_{x} = \|x_{s}^{nonl}-x_{s-1}^{nonl}\|_{2}/\|x_{s-1}^{nonl}\|_{2}<tol,\] \[err_{y} = \|y_{s}^{nonl}-y_{s-1}^{nonl}\|_{2}/\|y_{s-1}^{nonl}\|_{2}<tol,\]  
where \(tol=10 eps\). The scheme converges for perturbations \(\delta A\) of restricted size. It is possible that \(y\) converges while \(x\) does not converge. The nonlinear estimate of the higher term \(\Delta_{1}\) can be used to obtain nonlinear corrections of the singular value perturbations. Based on (40), a nonlinear correction of each singular value can be determined as  
\[\delta\sigma_{i}^{corr}=(|\delta D_{U1}|\Sigma_{n}+\Sigma_{n}|\delta D_{V}|+| \Delta_{1}|)_{ii}, \tag{66}\]  
so that the corresponding singular value perturbation is estimated as  
\[\delta\sigma_{i}^{nonl}=\sigma_{i}^{lin}+\delta\sigma_{i}^{corr}.\]  
Note that \(\sigma_{i}^{lin}=|\delta f_{ii}|\) is known only when the entries of the perturbation \(\delta A\) are known, and usually, this is not fulfilled in practice. Nevertheless, the nonlinear correction (66) can be useful in estimating the sensitivity of a given singular value. In Table 5, we present the number of iterations necessary to find the global bound \(x^{nonl}\) for the problem considered in Example 1 with perturbations \(\delta A=10^{c}\times A_{0},\ c=-10,-9,\ldots,-3\). In the last two columns of the table, we give the norm of the exact higher-order term \(\Delta\) and its approximation \(\|\Delta^{nonl}\|_{2}\) computed according to (58) and (59) (the approximation is given for the last iteration). In particular, for the perturbation \(\delta A=10^{-5}A_{0}\), the exact higher-order term \(\Delta\), found using (24) and (25), is \[|\Delta| = 10^{-7}\times\left[ \begin{array}{cccc}0.0045563&0.0006839&0.014619 5&0.0257942\\ 0.0012737&0.0008692&0.0080543&0.0003708\\ 0.0000993&0.0015257&0.0086067&0.0085598\\ 0.0019107&0.0007767&0.0258012&0.0170781\\ 0.0032494&0.0006701&0.0423789&0.2046100\\ 0.0064032&0.0004567&0.0384765&0.0071094\\ \end{array} \right],\] \[\|\Delta\|_{2} = 2.12746\times 10^{-8}.\]  
Implementing the described iterative procedure, after 10 iterations, we obtain the nonlinear bound  
\[\Delta^{nont} = 10^{-5}\times\left[ \begin{array}{cccc}0.0019716&0.0021343&0.004 9111&0.0047911\\ 0.0041273&0.0053422&0.0069174&0.0069309\\ 0.0189857&0.0186412&0.0222135&0.0179503\\ 0.8671799&0.8680880&0.8664831&0.8607933\\ \hline 0.5146400&0.5162827&0.5207565&0.2618103\\ 0.5146400&0.5162827&0.5207565&0.2618103\\ \end{array} \right],\] \[\|\Delta^{nont}\|_{2} = 2.16338\times 10^{-5}\]  
computed according to (58) and (59) on the base of the nonlinear bound \(x^{nont}\). The global bounds \(\|x^{nont}\|_{2}\) and \(\|y^{nont}\|_{2}\), found for different perturbations along with the values of \(\|x\|_{2}\) and \(\|y\|_{2}\), are shown in Table 6. The results confirm that the global estimates of \(x\) and \(y\) are close to the corresponding asymptotic estimates. In Figures 3 and 4, we show the convergence of the relative errors  
\[err_{x}=\|x_{s}^{nont}-x_{s-1}^{nont}\|/\|x_{s-1}^{nont}\|\]  
and  
\[err_{y}=\|y_{s}^{nont}-y_{s-1}^{nont}\|/\|y_{s-1}^{nont}\|,\]