## 9 Global Perturbation Bounds
### Iterative Procedure for Finding Global Bounds of \(x\) and \(y\)  
Global componentwise perturbation bounds of the matrices \(U\) and \(V\) can be found using nonlinear estimates of the matrices \(\delta W_{U}\) and \(\delta W_{V}\), determined by (15) and (16), respectively. Such estimates are found correcting the linear estimates of the perturbation parameters \(x_{k}=u_{i}^{T}\delta u_{j}\) and \(y_{\ell}=v_{i}^{T}\delta v_{j}\) on each iteration step. This approach is used in [9; 32] (Ch. 4), and [33] in the normwise perturbation analysis of invariant subspaces and is related to the solution of the nonlinear equation  
\[\mathbf{T}x=g-\phi(x),\]  
where \(\mathbf{T}\) is a linear operator corresponding to the asymptotic perturbation estimate and \(\phi(x)\) reflects the higher order terms corresponding to the nonlinear correction of the estimate. In finding the normwise global perturbation bounds, the above operator equation can be solved analytically using the contraction mapping theorem. Unfortunately, in the determining of componentwise estimates it is necessary to solve a system of complicated nonlinear equations and in such case the determining of analytical nonlinear bounds is difficult. That is why, in finding global bounds, we shall implement an iterative technique, similar to the one presented in [26]. Like the asymptotic estimates, the global bounds can be found only knowing the norm of \(\delta A\). This is performed approximating in the best possible way the higher-order terms (15) and (16), using on each step the approximations of the parameter vectors \(x\) and \(y\). We note that the iteration convergence is established only experimentally, and the derivation of a convergence proof is a matter for further investigation. Thus, the proposed iterative procedure should be considered only as an illustrative one. Consider the case of estimating the matrix \(\delta W_{U}\). It is convenient to substitute the terms containing the perturbations \(\delta u_{j}\) in (15) with the quantities  
\[\delta w_{uj}=U^{T}\delta u_{j},\ j=1,2,\ldots,n,\]  
which have the same magnitude as \(\delta u_{j}\). Since  
\[\delta u_{i}^{T}\delta u_{j}=\delta u_{i}^{T}UU^{T}\delta u_{j}=\delta w_{u_{i }}^{T}\delta w_{uj},\]  
the absolute value of the matrix \(\delta W_{U}\in\mathbb{R}^{m\times n}\) (15) can be bounded as  
\[|\delta W_{U}^{toml}| = |U^{T}\delta U_{1}|=[|\delta w_{u1}|,|\delta w_{u2}|,\ldots,| \delta w_{nn}|]\] \[\preceq |\delta Q_{U}|+|\delta D_{U}|+|\delta N_{U}|,\]  
where\[|\delta Q_{U}|:=\left[ \begin{array}{c}\delta Q_{U1}\\ \hline\delta Q_{U2}\end{array} \right]=\left[\begin{array}{c}\begin{array}{c} \includegraphics[height=142.26378 pt]{201}\\ \includegraphics[height=142.26378 pt]{202}\\ \includegraphics[height=142.26378 pt]{203}\\ \includegraphics[height=142.26378 pt]{204}\\ \includegraphics[height=142.26378 pt]{205}\\ \includegraphics[height=142.26378 pt]{206}\\ \includegraphics[height=142.26378 pt]{207}\\ \includegraphics[height=142.26378 pt]{208}\\ \includegraphics[height=142.26378 pt]{209}\\ \includegraphics[height=142.26378 pt]{209}\\ \includegraphics[height=142. The nonlinear correction of \(x_{(2)}\) is found from  
\[\delta x_{(2)} = \text{vec}(Z), \tag{62}\] \[Z = \|\Delta_{2}^{nonl}\|_{2}\times\left[ \begin{array}{cccc}1/\sigma_ {1}&1/\sigma_{2}&\ldots&1/\sigma_{n}\\ 1/\sigma_{1}&1/\sigma_{2}&\ldots&1/\sigma_{n}\\ \vdots&\vdots&\vdots&\vdots\\ 1/\sigma_{1}&1/\sigma_{2}&\ldots&1/\sigma_{n}\end{array} \right].\]  
and the total correction vector is determined from  
\[\delta x=\Omega_{1}^{T}\delta x_{(1)}+\Omega_{2}^{T}\delta x_{(2)}. \tag{63}\]  
Now, the nonlinear estimates of the vectors \(x\) and \(y\) are found from  
\[x^{nonl} = x^{lin}+\delta x, \tag{64}\] \[y^{nonl} = y^{lin}+\delta y. \tag{65}\]  
In this way, we obtain an iterative scheme for finding simultaneously nonlinear estimates of the coupled perturbation parameter vectors \(x\) and \(y\) involving the Equations (56)-(59) and (60)-(65).