## 3 The Contour Integral Method for Solving NEP
### A Contour Integral Method
\right]\text{and }\mathbf{B}_{1}=\left[ \begin{array}{ccc}\mathbf{A}_{1}&\cdots&\mathbf{A}_{q}\\ \vdots&\ddots&\vdots\\ \mathbf{A}_{q}&\cdots&\mathbf{A}_{2q-1}\end{array} \right] \tag{19}\]  
Further, defining the following matrices  
\[\mathbf{V}_{\mathrm{q}}=\left[ \begin{array}{c}\mathbf{V}\\ \vdots\\ \mathbf{V}\Lambda^{q-1}\end{array} \right]\in\mathbb{C}^{qp\times m}\text{ and }\mathbf{W}_{\mathrm{q}}^{\mathrm{H}}=\left[\mathbf{W}^{ \mathrm{H}}\mathbf{R},\ldots,\Lambda^{q-1}\mathbf{W}^{\mathrm{H}}\mathbf{R} \right]\in\mathbb{C}^{m\times qp} \tag{20}\]  
Then, one has factorizations for \(\mathbf{B}_{0}\) and \(\mathbf{B}_{1}\) by (18)  
\[\mathbf{B}_{0}=\mathbf{V}_{\mathrm{q}}\mathbf{W}_{\mathrm{q}}^{\mathrm{H}} \text{ and }\mathbf{B}_{1}=\mathbf{V}_{\mathrm{q}}\Lambda\mathbf{W}_{\mathrm{q}}^{ \mathrm{H}} \tag{21}\]  
Here, one needs an assumption which is fulfilled in the generic case (e.g., \(q\geq m\)), that is,  
\[\mathrm{rank}\big{(}\mathbf{V}_{\mathrm{q}}\big{)}=\mathrm{rank}\big{(} \mathbf{W}_{\mathrm{q}}^{\mathrm{H}}\big{)}=m \tag{22}\]  
Now the next step is to compute an economy-size singular value decomposition (SVD) of \(\mathbf{B}_{0}\)  
\[\mathbf{B}_{0}=\mathbf{V}_{\mathrm{q}}\mathbf{W}_{\mathrm{q}}^{\mathrm{H}}= \mathbf{V}_{0}\mathbf{\Sigma}_{0}\mathbf{W}_{0}^{\mathrm{H}},\ \mathbf{V}_{0}\in\mathbb{C}^{qp\times m},\ \mathbf{W}_{0}\in\mathbb{C}^{qp\times m},\ \mathbf{\Sigma}_{0}=\mathrm{diag}(\sigma_{1}, \ldots,\sigma_{m}) \tag{23}\]  
Since \(\mathrm{range}\ \big{(}\mathbf{V}_{\mathrm{q}}\big{)}=\mathrm{range}\ \big{(}\mathbf{V}_{0}\big{)}\), there exists an invertible matrix \(\mathbf{X}\in\mathbb{C}^{m\times m}\) such that \(\mathbf{V}_{\mathrm{q}}=\mathbf{V}_{0}\mathbf{X}\). Substituting the expression \(\mathbf{V}_{\mathrm{q}}\) into \(\mathbf{V}_{\mathrm{q}}\mathbf{W}_{\mathrm{q}}^{\mathrm{H}}=\mathbf{V}_{0} \mathbf{\Sigma}_{0}\mathbf{W}_{0}^{\mathrm{H}}\) of (23) yields  
\[\mathbf{W}_{\mathrm{q}}^{\mathrm{H}}=\mathbf{X}^{-1}\mathbf{\Sigma}_{0} \mathbf{W}_{0}^{\mathrm{H}} \tag{24}\]  
This relation can be used to remove \(\mathbf{W}_{\mathrm{q}}^{\mathrm{H}}\) from \(\mathbf{B}_{1}=\mathbf{V}_{\mathrm{q}}\Lambda\mathbf{W}_{\mathrm{q}}^{\mathrm{ H}}\), i.e.,  
\[\mathbf{B}_{1}=\mathbf{V}_{0}\mathbf{X}\Lambda\mathbf{X}^{-1}\mathbf{\Sigma}_ {0}\mathbf{W}_{0}^{\mathrm{H}} \tag{25}\]  
Until now, one has thus arrived at  
\[\mathbf{P}=\mathbf{V}_{0}^{\mathrm{H}}\mathbf{B}_{1}\mathbf{W}_{0}\mathbf{ \Sigma}_{0}^{-1}=\mathbf{X}\Lambda\mathbf{X}^{-1} \tag{26}\]  
showing that the eigenvalue decomposition of the computable matrix \(\mathbf{P}\) leads to matrices \(\Lambda\) and \(\mathbf{X}\). To summarize, the solution of the nonlinear eigenvalue problem (14) inside the given contour \(\partial\mathcal{C}\) can be reduced to the solution of the linear eigenvalue problem (26), hence the eigenvalues of the NEP can be obtained from the eigenvalues of \(\mathbf{P}\), and the corresponding right eigenvectors can be retrieved from the columns of \(\mathbf{V}_{\mathrm{q}}=\mathbf{V}_{0}\mathbf{X}\). A similar approach was proposed by [PERSON] et al. [21] using the Smith normal form to express \(\mathbf{T}^{-1}(z)\).