## 8 Results and Discussion
### Example 4  
We solve a complex Helmholtz equation:  
\[-\Delta u(x,y)+\sigma u(x,y)=f(x,y),\ \ 0<x<1,\ \ 0<y<1, \tag{54}\]  
with homogeneous boundary condition, where \(\sigma\) is a complex-valued impedance coefficient. After five-point discretization, Equation (54) becomes a complex linear system:  
\[(\mathbf{W}+i\mathbf{T})(\mathbf{w}+i\mathbf{t})=\mathbf{f}+i\mathbf{g}, \tag{55}\]  
where \(\mathbf{W},\mathbf{T}\in\mathbb{R}^{N\times N}\) are symmetric positive semi-definite with \(\mathbf{W}>\mathbf{0}\) positive, and \(i^{2}=-1\). Upon letting  
\[\mathbf{A}=\left[ \begin{array}{cc}\mathbf{W}&-\mathbf{T}\\ \mathbf{T}&\mathbf{W}\end{array} \right]\text{, }\mathbf{y}=\left[ \begin{array}{c} \mathbf{w}\\ \mathbf{t}\end{array} \right]\text{, }\mathbf{h}=\left[ \begin{array}{c} \mathbf{f}\\ \mathbf{g}\end{array} \right]\text{,} \tag{56}\]  
Equation (55) becomes  
\[\mathbf{A}\mathbf{x}=\mathbf{b}\text{, }\mathbf{x}\text{, }\mathbf{b}\in \mathbb{R}^{n}\text{, }\ \ \mathbf{A}\in\mathbb{R}^{n\times n}\text{,} \tag{57}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\alpha\) & \(\mathbf{1.25}\) & \(\mathbf{1.3}\) & \(\mathbf{1.4}\) & \(\mathbf{1.45}\) & \(\mathbf{1.5}\) & \(\mathbf{1.6}\) \\ \hline NI & 114 & 77 & 100 & 85 & 98 & 142 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Example 3, NI for A2-PROR with different values of \(\alpha\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\alpha\) & \(\mathbf{1.3}\) & \(\mathbf{1.4}\) & \(\mathbf{1.45}\) & \(\mathbf{1.5}\) & \(\mathbf{1.6}\) \\ \hline NI & 113 & 128 & 109 & 140 & 165 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Example 3, NI for A2-SOR with different values of \(\alpha\). where \(n=2N\). Equation (57) is a two-block linear system; in general, special splitting techniques are designed to effectively solve this sort linear system, for instance, the optimal two-block splitting iterative scheme [8], the generalized successive overrelaxation (GSOR) method [39]; the symmetric block triangular splitting (SBTS) iteration method [40], the Hermitian and skew-Hermitian splitting (HSS) iteration method [41], and the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method [42]. Suppose that \(\mathbf{K}=\mathbf{I}_{n_{0}}\otimes\mathbf{C}+\mathbf{C}\otimes\mathbf{I}_{n_ {0}}\) be the centered difference matrix approximation of the negative Laplacian operator in Equation (54); the mesh size is \(h=1/(n_{0}+1)\), and \(\mathbf{C}=\mathrm{tridiag}(-1,2,-1)/h^{2}\); then we have  
\[\left[\mathbf{K}+\frac{3-\sqrt{3}}{h}\mathbf{I}_{N}+i\Bigg{(}\mathbf{K}+\frac {3+\sqrt{3}}{h}\mathbf{I}_{N}\Bigg{)}\right]u=\mathbf{f}+i\mathbf{g},\]  
where \(N=n_{0}^{2}\) and \(n=2n_{0}^{2}\);  
\[\mathbf{W}=\mathbf{K}+\frac{3-\sqrt{3}}{h}\mathbf{I}_{N},\ \mathbf{T}= \mathbf{K}+\frac{3+\sqrt{3}}{h}\mathbf{I}_{N},\] \[\mathbf{f}=\frac{j}{h(j+1)^{2}},\ \mathbf{g}=-\frac{j}{h(j+1)^{2}},\ j=1, \ldots,N.\]  
We apply the AOR in Equation (43) with \(w=0.6\) and \(r\) being optimized by Algorithm 5. Table 15 compares the optimal value of \(r\), the number of steps (NI) and CPU obtained by AOR, under \(\varepsilon=10^{-4}\) and \(\mathbf{x}_{0}=\mathbf{0}\). We apply the AOR in Equation (43) with \(w=1.2\) and \(r\) being optimized by Algorithm 5 at every iteration steps. It is equivalent to A2-SOR. Table 16 compares NI and CPU under \(\varepsilon=10^{-4}\) and \(\mathbf{x}_{0}=\mathbf{0}\). Comparing to Table 15 the A2-SOR is saving CPU time and also with less number of NI. Under \(n_{0}=32\) (\(n=2048\)), \(\|\mathbf{r}\|/\|\mathbf{b}\|\leq\varepsilon=10^{-6}\), where \(\|\mathbf{b}\|=\sqrt{\|\mathbf{b}_{1}\|^{2}+\|\mathbf{b}_{2}\|^{2}}\), \(\mathbf{b}_{1}=\mathbf{f}+\mathbf{g}=\mathbf{0}\) and \(\mathbf{b}_{2}=\mathbf{g}\), we apply the A1-AOR with \(\alpha=1.95\), \(w=1.6\) and \(r\) being optimized by Algorithm 5 at every iteration steps to solve this problem. We compare NI obtained by different methods in Table 17, some of which is listed in [40] with HSS in [41], and MHSS in [42]. The GSOR was obtained from Table 1 in [39].