## 4 Equations for the Perturbation Parameters
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(c\) & \(\|\delta A\|_{2}\) & \(x_{k}=u_{t}^{T}\delta u_{j}\) & \(|x_{k}|\) & \(x_{k}^{lin}\) \\ \hline \(-10\) & \(2.0819813\times 10^{-9}\) & \(x_{1}=u_{t}^{T}\delta u_{1}\) & \(5.9651314\times 10^{-12}\) & \(1.1712418\times 10^{-10}\) \\  & & \(x_{2}=u_{t}^{T}\delta u_{1}\) & \(1.2358232\times 10^{-11}\) & \(8.5991735\times 10^{-11}\) \\  & & \(x_{3}=u_{t}^{T}\delta u_{1}\) & \(2.5843633\times 10^{-12}\) & \(8.1830915\times 10^{-11}\) \\  & & \(x_{4}=u_{t}^{T}\delta u_{1}\) & \(7.5119150\times 10^{-13}\) & \(8.1773996\times 10^{-11}\) \\ \cline{2-5}  & & \(x_{6}=u_{6}^{T}\delta u_{1}\) & \(1.9764954\times 10^{-11}\) & \(8.1773996\times 10^{-10}\) \\ \hline  & & \(x_{6}=u_{t}^{T}\delta u_{2}\) & \(2.5769606\times 10^{-11}\) & \(3.2351171\times 10^{-10}\) \\  & & \(x_{7}=u_{t}^{T}\delta u_{2}\) & \(6.7934677\times 10^{-12}\) & \(2.7156394\times 10^{-10}\) \\  & & \(x_{8}=u_{t}^{T}\delta u_{2}\) & \(7.1177269\times 10^{-11}\) & \(2.7093810\times 10^{-10}\) \\ \cline{2-5}  & & \(x_{9}=u_{6}^{T}\delta u_{2}\) & \(3.0802793\times 10^{-11}\) & \(2.7093810\times 10^{-10}\) \\ \hline  & & \(x_{10}=u_{t}^{T}\delta u_{3}\) & \(4.9634458\times 10^{-10}\) & \(1.6912007\times 10^{-9}\) \\  & & \(x_{11}=u_{t}^{T}\delta u_{3}\) & \(6.0967501\times 10^{-10}\) & \(1.6672173\times 10^{-9}\) \\ \cline{2-5}  & & \(x_{12}=u_{6}^{T}\delta u_{3}\) & \(6.6936869\times 10^{-10}\) & \(1.6672173\times 10^{-9}\) \\ \cline{2-5}  & & \(x_{13}=u_{t}^{T}\delta u_{4}\) & \(1.0685519\times 10^{-8}\) & \(1.1756467\times 10^{-7}\) \\  & & \(x_{14}=u_{6}^{T}\delta u_{4}\) & \(1.0010811\times 10^{-9}\) & \(1.1756467\times 10^{-7}\) \\ \hline \(-5\) & \(2.0819812\times 10^{-4}\) & \(x_{1}=u_{t}^{T}\delta u_{1}\) & \(5.9650702\times 10^{-7}\) & \(1.1712418\times 10^{-5}\) \\  & & \(x_{2}=u_{t}^{T}\delta u_{1}\) & \(1.2358210\times 10^{-6}\) & \(8.5991735\times 10^{-6}\) \\  & & \(x_{3}=u_{t}^{T}\delta u_{1}\) & \(2.5843445\times 10^{-7}\) & \(8.1830915\times 10^{-6}\) \\ \cline{2-5}  & & \(x_{4}=u_{t}^{T}\delta u_{1}\) & \(7.5103866\times 10^{-8}\) & \(8.1773996\times 10^{-6}\) \\  & & \(x_{5}=u_{t}^{T}\delta u_{1}\) & \(1.9765187\times 10^{-6}\) & \(8.1773996\times 10^{-6}\) \\ \hline  & & \(x_{6}=u_{t}^{T}\delta u_{2}\) & \(2.5769484\times 10^{-6}\) & \(3.2351171\times 10^{-5}\) \\  & & \(x_{7}=u_{t}^{T}\delta u_{2}\) & \(6.7937087\times 10^{-7}\) & \(2.7156394\times 10^{-5}\) \\  & & \(x_{8}=u_{t}^{T}\delta u_{2}\) & \(7.1177424\times 10^{-6}\) & \(2.7093809\times 10^{-5}\) \\ \cline{2-5}  & & \(x_{9}=u_{t}^{T}\delta u_{2}\) & \(3.0802821\times 10^{-6}\) & \(2.7093809\times 10^{-5}\) \\ \hline  & & \(x_{10}=u_{t}^{T}\delta u_{3}\) & \(4.9636511\times 10^{-5}\) & \(1.6912007\times 10^{-4}\) \\ \cline{2-5}  & & \(x_{11}=u_{t}^{T}\delta u_{3}\) & \(6.0971010\times 10^{-5}\) & \(1.6672173\times 10^{-4}\) \\ \cline{2-5}  & & \(x_{12}=u_{t}^{T}\delta u_{3}\) & \(6.6939951\times 10^{-5}\) & \(1.6672173\times 10^{-4}\) \\ \hline  & & \(x_{13}=u_{t}^{T}\delta u_{4}\) & \(1.0673940\times 10^{-3}\) & \(1.1756467\times 10^{-2}\) \\  & & \(x_{14}=u_{t}^{T}\delta u_{4}\) & \(1.0014883\times 10^{-4}\) & \(1.1756467\times 10^{-2}\) \\ \hline \hline \end{tabular} Note: The elements \(x_{k}\) obtained from Equation (38) are highlighted in yellow boxes.  
\end{table} 
Table 2: Exact perturbation parameters \(y_{\ell}\) related to the matrix \(\delta W_{V}\) and their linear estimates.