## 7 Asymptotic Bounds on the Perturbations of \(U_{1}\) and \(V\)
: _The asymptotic bound on the matrix \(|\delta U_{1}|\) is given by_  
\[|\delta U_{1}|\preceq|U||U^{T}\delta U_{1}|\preceq\delta U_{1}^{lin}=|U|\delta W_ {U}^{lin}, \tag{42}\]  
_where_  
\[\delta W_{U}^{lin}=\left[\begin{array}{ccccc}\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{ -2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{-2.eps}\\ \includegraphics[width=142.26378 pt]{-2.eps}&\includegraphics[width=142.26378 pt]{-2. For the matrix \(A\) of Example 1, we obtain that the absolute values of the exact changes of the entries of the matrix \(U_{1}\) for the perturbation \(\delta A=10^{-5}A_{0}\) satisfy  
\[|\delta U_{1}|=10^{-3}\times\left[ \begin{array}{cccc}0.0020130&0.0013238&0.03304 02&0.2407168\\ 0.0009548&0.0070093&0.0356216&0.9425346\\ 0.0006562&0.0004107&0.0576633&0.1185295\\ 0.0000996&0.0028839&0.0020370&0.1517173\\ 0.0004584&0.0024637&0.0316821&0.3649600\\ 0.0004974&0.0014734&0.0630623&0.1877213\end{array} \right]\]  
and their asymptotic componentwise estimates found by using (42) are  
\[\delta U_{1}^{lin}=10^{-2}\times\left[ \begin{array}{cccc}0.0018494&0.0052303& 0.0215558&0.9587781\\ 0.0012398&0.0044262&0.0221410&1.4004629\\ 0.0014125&0.0045112&0.0219559&0.5375840\\ 0.0017421&0.0041696&0.0159680&0.5367878\\ 0.0015062&0.0033948&0.0117823&0.6368282\\ 0.0016843&0.0050287&0.0180170&0.9570612\end{array} \right].\]  
Also, for the same \(\delta A\), we have that  
\[|\delta V|=10^{-4}\times\left[ \begin{array}{cccc}0.0188135&0.0293793&0.262196 7&0.0365476\\ 0.0348050&0.0032323&0.0065525&0.2572914\\ 0.0153096&0.0121733&0.0836556&0.0994065\\ 0.0036567&0.0106224&0.0446413&0.0002219\end{array} \right]\]  
and, according to (43),  
\[\delta V^{lin}=10^{-3}\times\left[