## 2 Spectral Analysis
### Numerical Evidence: Preconditioning, Visualizations, GMRES Iterations  
In the present subsection, we consider the preconditioned matrices. More in detail, Tables 1-4 concern the measure of the clustering at 1 with radius \(\epsilon=0.1,0.01\), for \(\gamma=0.5\), 0.8, 1, 1.5, for \(\alpha=1.2,1.4,1.6,1.8\), for various matrix-dimensions \(n^{2}=2^{8},2^{10},2^{12}\). As can be seen, the number \(N_{o}(\epsilon)=N_{o}(\epsilon,n)\) increases moderately with \(n\), but the percentage of the number of outliers with respect to the matrix-size tends to zero fast and this is in agreement with the forecast of Theorem 4, at least for \(\gamma<1\): the situation is indeed better than the theoretical predictions because even when the condition \(\gamma<1\) is violated, we still observe a clustering at 1 and this is not a surprise given the comments after Theorem 4. \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{\(\boldsymbol{\mu(x,y)=1/(x+iy)^{\gamma}}\), \(\boldsymbol{\gamma=0.5}\)} \\ \hline \(n\) & \(N_{o}\)**(0.1)** & \(\boldsymbol{Percentage}\) & \(N_{o}\)**(0.01)** & \(\boldsymbol{Percentage}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.2\)} \\ \(2^{4}\) & 2 & \(7.812500\times 10^{-1}\) & 227 & \(8.867188\times 10^{1}\) \\ \(2^{5}\) & 6 & \(5.859375\times 10^{-1}\) & 302 & \(2.949219\times 10^{1}\) \\ \(2^{6}\) & 16 & \(3.906250\times 10^{-1}\) & 307 & \(7.495117\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.4\)} \\ \(2^{4}\) & 1 & \(3.906250\times 10^{-1}\) & 89 & \(3.476562\times 10^{1}\) \\ \(2^{5}\) & 5 & \(4.882812\times 10^{-1}\) & 99 & \(9.667969\times 10^{0}\) \\ \(2^{6}\) & 13 & \(3.173828\times 10^{-1}\) & 154 & \(3.759766\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.6\)} \\ \(2^{4}\) & 0 & 0 & 35 & \(1.367188\times 10^{1}\) \\ \(2^{5}\) & 3 & \(2.929688\times 10^{-1}\) & 60 & \(5.859375\times 10^{0}\) \\ \(2^{6}\) & 8 & \(1.953125\times 10^{-1}\) & 112 & \(2.734375\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.8\)} \\ \(2^{4}\) & 0 & 0 & 20 & \(7.812500\times 10^{0}\) \\ \(2^{5}\) & 0 & 0 & 38 & \(3.710938\times 10^{0}\) \\ \(2^{6}\) & 0 & 0 & 73 & \(1.782227\times 10^{0}\) \\ \hline \end{tabular}
\end{table} 
Table 1: 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 \multicolumn{5}{c}{\(\boldsymbol{\mu(x,y)=1/(x+iy)^{\gamma}}\), \(\boldsymbol{\gamma=0.8}\)} \\ \hline \(n\) & \(N_{o}\)**(0.1)** & \(\boldsymbol{Percentage}\) & \(N_{o}\)**(0.01)** & \(\boldsymbol{Percentage}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.2\)} \\ \(2^{4}\) & 3 & \(1.171875\times 10^{0}\) & 233 & \(9.101562\times 10^{1}\) \\ \(2^{5}\) & 7 & \(6.835938\times 10^{-1}\) & 395 & \(3.857422\times 10^{1}\) \\ \(2^{6}\) & 16 & \(3.906250\times 10^{-1}\) & 456 & \(1.113281\times 10^{1}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.4\)} \\ \(2^{4}\) & 2 & \(7.812500\times 10^{-1}\) & 115 & \(4.492188\times 10^{1}\) \\ \(2^{5}\) & 6 & \(5.859375\times 10^{-1}\) & 131 & \(1.279297\times 10^{1}\) \\ \(2^{6}\) & 15 & \(3.662109\times 10^{-1}\) & 182 & \(4.443359\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.6\)} \\ \(2^{4}\) & 0 & 0 & 46 & \(1.796875\times 10^{1}\) \\ \(2^{5}\) & 3 & \(2.929688\times 10^{-1}\) & 66 & \(6.445312\times 10^{0}\) \\ \(2^{6}\) & 8 & \(1.953125\times 10^{-1}\) & 116 & \(2.832031\times 10^{0}\) \\ \hline \multicolumn{5}{c}{\(\alpha=1.8\)} \\ \(2^{4}\) & 0 & 0 & 20 & \(7.812500\times 10^{0}\) \\ \(2^{5}\) & 0 & 0 & 39 & \(3.808594\times 10^{0}\) \\ \(2^{6}\) & 0 & 0 & 75 & \(1.831055\times 10^{0}\) \\ \hline \end{tabular}
\end{table} 
Table 2: 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 \(n^{2}\).