## 2 Spectral Analysis
### Numerical Evidence: Preconditioning, Visualizations, GMRES Iterations
The clustering in the preconditioning setting is also visualized in Figures 5-8, while Table 5 accounts for the fact that the localization around \(1\) is very good since we do not have large outliers. The moderate size of the outliers indicates that the preconditioned GMRES is expected to be optimal and robust with respect to all the involved parameters. The latter is evident in Table 6 with a slight increase in the number of iterations when \(\gamma\) increases, so that the number and the magnitude of the outliers are slightly larger. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\mu(x,y)=1/(x+\mathrm{i}y)^{\gamma}\), \(\gamma=1\)} \\ \hline \(n\) & \(N_{o}(0.1)\) & \(\boldsymbol{Percentage}\) & \(N_{o}(0.01)\) & \(\boldsymbol{Percentage}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.2\)} \\ \(2^{4}\) & \(20\) & \(7.812500\times 10^{0}\) & \(237\) & \(9.257812\times 10^{1}\) \\ \(2^{5}\) & \(33\) & \(3.222656\times 10^{0}\) & \(547\) & \(5.341797\times 10^{1}\) \\ \(2^{6}\) & \(55\) & \(1.342773\times 10^{0}\) & \(923\) & \(2.253418\times 10^{1}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.4\)} \\ \(2^{4}\) & \(9\) & \(3.515625\times 10^{0}\) & \(154\) & \(6.015625\times 10^{1}\) \\ \(2^{5}\) & \(13\) & \(1.269531\times 10^{0}\) & \(246\) & \(2.402344\times 10^{1}\) \\ \(2^{6}\) & \(24\) & \(5.859375\times 10^{-1}\) & \(367\) & \(8.959961\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.6\)} \\ \(2^{4}\) & \(3\) & \(1.171875\times 10^{0}\) & \(78\) & \(3.046875\times 10^{1}\) \\ \(2^{5}\) & \(5\) & \(4.882812\times 10^{-1}\) & \(117\) & \(1.142578\times 10^{1}\) \\ \(2^{6}\) & \(11\) & \(2.685547\times 10^{-1}\) & \(181\) & \(4.418945\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.8\)} \\ \(2^{4}\) & \(1\) & \(3.906250\times 10^{-1}\) & \(42\) & \(1.640625\times 10^{1}\) \\ \(2^{5}\) & \(1\) & \(9.765625\times 10^{-2}\) & \(61\) & \(5.957031\times 10^{0}\) \\ \(2^{6}\) & \(1\) & \(2.441406\times 10^{-2}\) & \(97\) & \(2.368164\times 10^{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of outliers \(N_{o}(\epsilon)\) with respect to a neighborhood of \(1\) of radius \(\epsilon=0.1\) or \(\epsilon=0.01\) and related percentage for increasing dimension \(n^{2}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\mu(x,y)=1/(x+\mathrm{i}y)^{\gamma}\), \(\gamma=1\)} \\ \hline \(n\) & \(N_{o}(0.1)\) & \(\boldsymbol{Percentage}\) & \(N_{o}(0.01)\) & \(\boldsymbol{Percentage}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.2\)} \\ \(2^{4}\) & \(8\) & \(3.125000\times 10^{0}\) & \(235\) & \(9.179688\times 10^{1}\) \\ \(2^{5}\) & \(11\) & \(1.074219\times 10^{0}\) & \(450\) & \(4.394531\times 10^{1}\) \\ \(2^{6}\) & \(22\) & \(5.371094\times 10^{-1}\) & \(588\) & \(1.435547\times 10^{1}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.4\)} \\ \(2^{4}\) & \(2\) & \(7.812500\times 10^{-1}\) & \(128\) & \(50\) \\ \(2^{5}\) & \(6\) & \(5.859375\times 10^{-1}\) & \(164\) & \(1.601562\times 10^{1}\) \\ \(2^{6}\) & \(15\) & \(3.662109\times 10^{-1}\) & \(217\) & \(5.297852\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.6\)} \\ \(2^{4}\) & \(0\) & \(0\) & \(58\) & \(2.265625\times 10^{1}\) \\ \(2^{5}\) & \(2\) & \(1.953125\times 10^{-1}\) & \(76\) & \(7.421875\times 10^{0}\) \\ \(2^{6}\) & \(8\) & \(1.953125\times 10^{-1}\) & \(123\) & \(3.002930\times 10^{0}\) \\ \(\alpha=1.8\) & & & & \\ \hline \multicolumn{5}{c}{\(\alpha=1.8\)} \\ \(2^{4}\) & \(0\) & \(0\) & \(27\) & \(1.054688\times 10^{1}\) \\ \(2^{5}\) & \(0\) & \(0\) & \(43\) & \(4.199219\times 10^{0}\) \\ \(2^{6}\) & \(0\) & \(0\) & \(78\) & \(1.904297\times 10^{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Number of outliers \(N_{o}(\epsilon)\) with respect to a neighborhood of \(1\) of radius \(\epsilon=0.1\) or \(\epsilon=0.01\) and related percentage for increasing dimension \(n^{2}\). Figure 5: Eigenvalues of the preconditioned matrix of size \(n^{2}=2^{12}\) for \(\gamma=0.5\) and \(\alpha=\{1.2,1.4,1.6,1.8\}\), respectively.