## 5 Case Studies
### All-Island Irish Transmission System
#### 5.1.3 Sparse Algorithms
The most relevant issue is that the eigenvalues obtained with z-PARES are not the most important ones for the stability of the system, which means that critical eigenvalues are missed. This issue occurs despite the defined search contour being reasonable. Of course, there may be some region for which the critical eigenvalues are captured, but this can not be known _a priori_. Regarding the simulation time, the method for the AIITS is faster than SLEPc's Krylov subspace methods for the LEP, but slower for the GEP. Figure 4 shows the search contour and the location of the characteristic roots that are found by z-PARES for the LEP. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Library** & **ARPACK** & **SLEPc** & **SLEPc** \\ \hline Method & IR-Arnoldi & ERD-Arnoldi & \\ Spectrum & 50 LRP & 50 TRP & 50 TRP \\ Transform & - & Shift \& invert & Shift \& invert \\  & & \(\sigma=-0.01\) & \(\sigma=-0.01\) \\ \hline Time [s] & 76.96 & 17.84 & 16.58 \\ Found & 26 eigs. & 54 eigs. & 55 eigs. \\ \hline LRP eigs. & \(-0.0000\) & \(0.0000\) & \(0.0000\) \\  & \(-0.0869\) & \(-0.0869\) & \(-0.0869\) \\  & \(-0.1276\pm j\,0.1706\) & \(-0.1276\pm j\,0.1706\) & \(-0.1276\pm j\,0.1706\) \\  & \(-0.1322\pm j\,0.4353\) & \(-0.1322\pm j\,0.4353\) & \(-0.1322\pm j\,0.4353\) \\  & \(-0.1615\pm j\,0.2689\) & \(-0.1376\) & \(-0.1376\) \\  & \(-0.1809\pm j\,0.2859\) & \(-0.1382\) & \(-0.1382\) \\  & \(-0.2042\pm j\,0.3935\) & \(-0.1386\) & \(-0.1386\) \\  & \(-0.2172\pm j\,0.2646\) & \(-0.1390\) & \(-0.1390\) \\  & \(-0.2335\pm j\,0.3546\) & \(-0.1391\) & \(-0.1391\) \\  & \(-0.2344\pm j\,0.3644\) & \(-0.1393\) & \(-0.1393\) \\  & \(-0.2503\pm j\,0.4363\) & \(-0.1394\) & \(-0.1394\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: AIITS: Krylov subspace methods, LEP. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Library** & \multicolumn{2}{c}{**SLEPc**} \\ \hline Method & ERD-Arnoldi & Krylov-Schur \\ Spectrum & 50 TRP & 50 TRP \\ Transform & Shift \& invert & Shift \& invert \\  & \(\sigma=-0.01\) & \(\sigma=-0.01\) \\ \hline Time [s] & 8.93 & 7.64 \\ Found & 51 eigs. & 53 eigs. \\ \hline LRP eigs. & 0.0000 & 0.0000 \\  & \(-0.0869\) & \(-0.0869\) \\  & \(-0.1276\pm\) / 0.1706 & \(-0.1276\pm\) / 0.1706 \\  & \(-0.1322\pm\) / 0.4353 & \(-0.1322\pm\) / 0.4353 \\  & \(-0.1376\) & \(-0.1376\) \\  & \(-0.1382\) & \(-0.1382\) \\  & \(-0.1386\) & \(-0.1386\) \\  & \(-0.1390\) & \(-0.1390\) \\  & \(-0.1391\) & \(-0.1391\) \\  & \(-0.1393\) & \(-0.1393\) \\  & \(-0.1394\) & \(-0.1394\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: AIITS: Krylov subspace methods, GEP. Figure 4: AIITS: root loci obtained with z-PARES, LEP. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Library** & \multicolumn{2}{c}{**z-PARES**} \\ \hline Method & \multicolumn{2}{c}{CI-RR} \\ Spectrum & \multicolumn{2}{c}{\(\gamma=(-0.01,4),\rho=8\)} \\ \hline Problem & LEP & GEP \\ Time [s] & 10.81 & 17.10 \\ Found & 49 eigs. & 52 eigs. \\ \hline LRP eigs. & \(-0.3041+\) / 4.1425 & \(-0.3040+\) / 4.1429 \\  & \(-0.3720+\) / 4.7773 & \(-0.3715+\) / 4.7774 \\  & \(-0.3945+\) / 4.3121 & \(-0.3947+\) / 4.3122 \\  & \(-0.4184+\) / 3.6794 & \(-0.4187\pm\) / 3.6794 \\  & \(-0.4866+\) / 5.0405 & \(-0.4865+\) / 5.0405 \\  & \(-0.5011+\) / 4.1276 & \(-0.5007+\) / 4.1274 \\  & \(-0.5022+\) / 4.4417 & \(-0.5018+\) / 4.4417 \\  & \(-0.5077+\) / 5.8727 & \(-0.5097+\) / 5.8747 \\  & \(-0.5555+\) / 5.3444 & \(-0.5542+\) / 5.3436 \\  & \(-0.6765+\) / 6.3426 & \(-0.6761+\) / 6.3412 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: AIITS: contour integration method, LEP and GEP.