## 6 Conclusions  
The paper provides a comprehensive comparison of algorithms and open-source software tools for the numerical solution of the non-Hermitian eigenvalue problems that arise in power systems.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Library** & \multicolumn{3}{c}{**z-PARES**} \\ \hline Problem & \multicolumn{3}{c}{GEP} \\ Method & \multicolumn{3}{c}{CI-RR} \\ Spectrum & \multicolumn{3}{c}{\(\gamma=(-0.01,4),\rho=8\)} \\ \hline Transform & - & Invert & Inverted Cayley \\ \hline Time [s] & 364.85 & 350.82 & 337.43 \\ Found & 349 eigs. & 297 eigs. & 349 eigs. \\ \hline \hline \end{tabular}
\end{table}
Table 14: ENTSO-E: impact of spectral transforms of the CI-RR method.  
In particular, the paper considers state-of-art implementations of methods that are based on Schur decomposition, vector iteration, Krylov subspace, and contour integration.  
The most robust and reliable method for the solution of the LEP for small to medium size problems is the QR algorithm with LAPACK. For large-scale problems, which is the case of real-world power systems, the only option is to use software tools that exploit matrix sparsity. While different options exist, such methods are less mature than the QR algorithm. The simulation results discussed in this paper recommend that, among the available options, the most promising for power systems are the parallel versions of the Krylov-Schur method by SLEPc and CI-RR method by z-PARES. In particular, SLEPc's Krylov-Schur provides the best accuracy, while z-PARES's CI-RR was the only one that was able to tackle the 146,166 \(\times\) 146,166 GEP of the European Network of Transmission System Operators for Electricity (ENTSO-E).  
**Author Contributions:** Formal analysis, G.T., I.D., M.L. and F.M.; Methodology, G.T., I.D., M.L. and F.M.; Writing--original draft, G.T., I.D., M.L. and F.M. The authors claim to have contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by: Science Foundation Ireland (Grant No. SFI/15/IA/3074), by funding [PERSON], [PERSON], [PERSON], and [PERSON]; [PERSON] and [PERSON] are also supported by European Union's Horizon 2020 research and innovation programme (Grant No. 883710).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Notation**  
\begin{tabular}{l l}
**a** & vector \\
**A** & matrix \\
**A**\({}^{\rm H}\) & matrix conjugate transpose (Hermitian) \\
**C** & complex numbers \\
**H** & Hessenberg matrix \\
**I**\({}_{r}\) & identity matrix of dimensions \(r\times r\) \\
**J** & unit imaginary number \\
**J** & Hankel matrix \\
**R** & upper triangular matrix of QR decomposition \\
**R** & real numbers \\
**s** & complex Laplace variable \\
**T** & upper triangular matrix of Schur decomposition \\
**u** & right eigenvector \\
**w** & left eigenvector \\
**x** & vector of state variables \\
**y** & vector of algebraic variables \\
**z** & spectral transform \\ \(\gamma\) & center of circular contour in the complex plane \\ \(\lambda\) & eigenvalue \\ \(\mu\) & moment associated to matrix pencil \\ \(\
u\) & spectral anti-shift \\ \(\rho\) & radius of circular contour in the complex plane \\ \(\sigma\) & spectral shift \\ \(\psi\) & contour integrals associated to matrix pencil \\
**0\({}_{n,m}\)** & zero matrix of dimensions \(n\times m\) \\ \end{tabular}