# Improved Convergence Speed of a DCD-Based Algorithm for Sparse Solutions
## 3 Proposed Leading-Cyclic DCD Algorithm
Among these values of \(N_{u_{\textit{leading}}}\), the one that allows the proposed algorithm to achieve the fastest convergence to the steady-state level is chosen as the optimal \(N_{u_{\textit{leading}}}\). Figure 5 shows the optimal \(N_{u_{\textit{leading}}}\) in the leading-cyclic DCD algorithm for matrices with different sparsities \(\gamma\). The misalignments compared to the number of successful iterations for large system matrices with condition numbers (\(\geq\)100) and sparse solutions are shown in Figures 6-11. By comparing the results from Figures 6 and 7, we can see that the lower the condition number of the system matrix is, the faster the convergence speed and the lower the steady-state level of these DCD algorithms. From these figures, we can see that when \(\gamma\leq 1/8\), the leading DCD algorithm has a significantly faster convergence speed and a lower steady-state misalignment than the cyclic DCD algorithm. The leading-cyclic DCD algorithm has approximately the same steady-state level as the leading DCD algorithm. As \(\gamma\) increases, the cyclic DCD algorithm gradually outperforms the leading DCD algorithm. The leading-cyclic DCD algorithm with a given \(N_{u_{\textit{leading}}}\) (shown in Figure 5) has an increased convergence speed and a lower steady-state misalignment compared to the leading DCD algorithm. When \(\gamma=2/8\) increases to \(\gamma=6/8\), the leading-cyclic DCD algorithm exhibits an insignificantly increased convergence speed compared to the cyclic DCD algorithm. Figures 8-11 show the misalignments of the DCD algorithms with conditional numbers of the system matrix in the interval [300, 400]. We can see that when \(\xi=Q/U\) increases, the convergence speed of the DCD algorithms decreases. Figure 5: Optimal \(N_{u_{\textit{leading}}}\) vs. sparsity. According to the numerical results above, we can see that the leading-cyclic DCD algorithm is an effective approach for solving a sparse system with a large number of updates. For example, in code division multiple access (CDMA)with a spreading factor \(Q\), there are approximately half of the users are active in the \(U\) user system. The columns of \(\mathbf{A}\) are represented as spreading sequences. The matrix \(\mathbf{R}\) is represented as a correlation matrix of the spreading sequence. Figure 12 shows the misalignments of the RLS algorithm and \(\text{DCD}_{leading-cyclic}\)-RLS (forgetting factor \(\lambda=0.96\), the length of the filter \(U=16\), and \(\gamma=1/2\)). When system noise energy increases by 10 times between 1000 iterations and 1100 iterations, the \(\text{DCD}_{leading-cyclic}\)-RLS maintains a low error rate. The proposed algorithm-based RLS provides a lower error level and better tracking performance than the RLS algorithm. Figure 6: Misalignments of the DCD algorithms for different sparse solutions with the condition numbers in the range of [100, 200] and sparse solutions; \(U=64\), \(M_{b}=15\), \(Q=81\), \(\xi=1.2656\). Figure 7: Misalignments of the DCD algorithms for different sparse solutions with the condition numbers in the range of [500, 600]; \(U=64\), \(M_{b}=15\), \(Q=71\), \(\xi=1.109\). Figure 8: Misalignments of the DCD algorithms for different sparse solutions with the condition numbers in the range of \([300,400]\); \(U=32\), \(M_{b}=15\), \(Q=37\), \(\xi=1.156\). Figure 10: Misalignments of the DCD algorithms for different sparse solutions with the condition numbers in the range of \([300,400]\); \(U=96\), \(M_{b}=15\), \(Q=113\), \(\xi=1.177\). Figure 9: Misalignments of the DCD algorithms for different sparse solutions with the condition numbers in the range of \([300,400]\); \(U=64\), \(M_{b}=15\), \(Q=75\), \(\xi=1.172\).