## 9 Global Perturbation Bounds
### Iterative Procedure for Finding Global Bounds of \(x\) and \(y\)
\begin{table}
\begin{tabular}{c c c c c} \hline \(\epsilon\) & \(\|x\|_{2}\) & \(\|x^{nont}\|_{2}\) & \(\|y\|_{2}\) & \(\|y^{nont}\|_{2}\) \\ \hline \(-10\) & \(1.0782203\times 10^{-8}\) & \(1.6628799\times 10^{-7}\) & \(2.8118399\times 10^{-10}\) & \(1.7511069\times 10^{-9}\) \\ \(-9\) & \(1.0782176\times 10^{-7}\) & \(1.6628817\times 10^{-6}\) & \(2.8118379\times 10^{-9}\) & \(1.7511072\times 10^{-8}\) \\ \(-8\) & \(1.0782167\times 10^{-6}\) & \(1.6628999\times 10^{-5}\) & \(2.8118393\times 10^{-8}\) & \(1.7511098\times 10^{-7}\) \\ \(-7\) & \(1.0782065\times 10^{-5}\) & \(1.6630816\times 10^{-4}\) & \(2.8118405\times 10^{-7}\) & \(1.7511358\times 10^{-6}\) \\ \(-6\) & \(1.0781037\times 10^{-4}\) & \(1.6649057\times 10^{-3}\) & \(2.8118536\times 10^{-6}\) & \(1.7513967\times 10^{-5}\) \\ \(-5\) & \(1.0770771\times 10^{-3}\) & \(1.6837998\times 10^{-2}\) & \(2.81198440\times 10^{-5}\) & \(1.7540888\times 10^{-4}\) \\ \(-4\) & \(1.0668478\times 10^{-2}\) & \(1.9743124\times 10^{-1}\) & \(2.8132907\times 10^{-4}\) & \(1.7960922\times 10^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Global bounds of \(x\) and \(y\). \begin{table}
\begin{tabular}{c c c c c} \hline \(\epsilon\) & \(\|\delta A\|_{F}\) & **Number of Iterations** & \(\|\Delta\|_{2}\) & \(\|\boldsymbol{\Delta}^{nont}\|_{2}\) \\ \hline \(-10\) & \(2.08198\times 10^{-9}\) & 4 & \(2.12806\times 10^{-18}\) & \(2.09530\times 10^{-15}\) \\ \(-9\) & \(2.08198\times 10^{-8}\) & 4 & \(2.12806\times 10^{-16}\) & \(2.09531\times 10^{-13}\) \\ \(-8\) & \(2.08198\times 10^{-7}\) & 5 & \(2.12806\times 10^{-14}\) & \(2.09537\times 10^{-11}\) \\ \(-7\) & \(2.08198\times 10^{-6}\) & 5 & \(2.12806\times 10^{-12}\) & \(2.09595\times 10^{-9}\) \\ \(-6\) & \(2.08198\times 10^{-5}\) & 7 & \(2.12800\times 10^{-10}\) & \(2.10185\times 10^{-7}\) \\ \(-5\) & \(2.08198\times 10^{-4}\) & 10 & \(2.12746\times 10^{-8}\) & \(2.16338\times 10^{-5}\) \\ \(-4\) & \(2.08198\times 10^{-3}\) & 29 & \(2.12189\times 10^{-6}\) & \(3.23896\times 10^{-3}\) \\ \(-3\) & \(2.08198\times 10^{-2}\) & No convergence & - & - \\ \hline \end{tabular}
\end{table} 
Table 5: Convergence of the global bounds and higher-order terms. respectively, at step \(s\) of the iterative process for different perturbations \(\delta A=10^{c}\times A_{0}\). (The value of the relative error on each step is shown by a circle.) As it is seen from the figures, for the given example the convergence of \(y\) is close to the convergence of \(x\). For all values of \(c\), except \(c=-3\), the relative error begins with values \(err_{x},\ err_{y}>>1\) and ends with values \(err_{x},\ err_{y}<10^{-13}\). With the increasing in the perturbation size, the convergence becomes slower and for \(c=-3\) (\(\|\delta A\|_{2}=2.08198\times 10^{-2}\)), the iteration diverges. This demonstrates the restricted usefulness of the nonlinear estimates, which is valid only for limited perturbation magnitudes. In Table 7, we give normwise perturbation bounds of the singular values along with the actual singular value perturbations and their global bounds found for two perturbations of \(A\) under the assumption that the linear bounds of all singular values are known. As can be seen from the table, the nonlinear estimates of the singu  
Figure 4: Relative errors in the iterative determining of the global bounds of \(y\). Figure 3: Relative errors in the iterative determining of the global bounds of \(x\).