## 5 Numerical Examples
### Example 3  
We consider an over-determined Hilbert linear problem:  
\[A_{ij}=\frac{1}{i+j-1},\ \ 1\leq i\leq q,\ \ 1\leq j\leq n,\] \[b_{i}=\sum_{j=1}^{n}\frac{1}{i+j-1}x_{j},\ \ 1\leq i\leq q, \tag{134}\]  
where we fix \(q=6\) and \(n=5\), and the exact solutions \(x_{j}=1/j,\ 1\leq j\leq n\) are given. The DOA with \(m=4\) and \(\varepsilon=10^{-13}\) can solve this problem very fast, starting from the initial guesses \(x_{1}^{0}=x_{2}^{0}=x_{3}^{0}=x_{4}^{0}=x_{5}^{0}=0\), in only four steps, as shown in Figure 2a. Furthermore, we can find a very accurate solution, as shown in Figure 2b with the maximum error (ME) being \(8.91\times 10^{-12}\). If we consider an under-determined system with \(q=5\) and \(n=6\), the DOA with \(m=3\) is still workable with ME = \(2.22\times 10^{-8}\). For least-squares problems, QR is a famous method to directly solve Equation (2). This process is shown in Equations (9)-(11). However, we find that ME = \(1.05\times 10^{-6}\) obtained by QR is less accurate than that obtained via the DOA in the Hilbert least-squares problem with \(q=6\) and \(n=5\). We have checked the accuracy of the orthogonality in Equation (10) by  
\[\|\mathbf{Q}^{\mathrm{T}}\mathbf{Q}-\mathbf{I}_{n}\|,\]  
which is on the order of \(8.65\times 10^{-12}\). The Hilbert matrix is known as a notorious example of a highly ill-conditioned matrix. For this problem, the condition number is \(2.5\times 10^{5}\). Then, in the floating point, \(\mathbf{b}\) has a rounding error on the the order of \(10^{-16}\). The algorithm introduces further floating point errors which are of the same order of magnitude. Perturbation theory for the least-squares problem [38] shows that one can only expect to find a solution that is approximately within \(2\times 10^{(5-16)}=2\times 10^{-11}\) of the exact solution. The error \(\|\mathbf{Q}^{\mathrm{T}}\mathbf{Q}-\mathbf{I}_{n}\|=8.65\times 10^{-12}\) is within this range; however, the ME = \(1.05\times 10^{-6}\) obtained by QR is not within this range. The reason for this is that when we use Equation (11) to find the solution in the QR method, very small singular values appearing in the denominator will enlarge the rounding error and the error from \(\mathbf{Q}\); for example, the last two singular values \(4.803\times 10^{-4}\) and \(1.734\times 10^{-5}\), upon dividing by those two small values sequentially, lead to ME = \(1.05\times 10^{-6}\), which is not within the range of \(2\times 10^{-11}\). In contrast, the proposed DOA method with an error of \(8.91\times 10^{-12}\) is within this range. For highly ill-conditionedleast-squares problems, the QR method loses some accuracy, and even \(\mathbf{Q}\) well satisfies Equation (10) within the range of \(2\times 10^{-11}\). We raise \(q\) to 10, and Table 1 lists the computed ME for different values of \(n\). We also list the error \(\|\mathbf{Q}^{\mathbf{T}}\mathbf{Q}-\mathbf{I}_{n}\|\) for the QR method, which is worse when \(n\) is increased. Based on Equations (88)-(90), this problem is solved by SVD. The DOA is competitive with SVD, and for all cases, it is slightly more accurate than SVD. To investigate the influence of \(m\) on the DOA, Table 2 compares the ME and the iteration number (IN). For ill-posed Hilbert problems, there exists the best value \(m=3\). \begin{table}
\begin{tabular}{c c c c c} \hline \(n\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) & \(\mathbf{5}\) \\ \hline \(\|\mathbf{Q}^{\mathbf{T}}\mathbf{Q}-\mathbf{I}_{n}\|\) & \(1.63\times 10^{-15}\) & \(2.48\times 10^{-14}\) & \(3.11\times 10^{-13}\) & \(5.24\times 10^{-12}\) \\ \hline QR & \(8.33\times 10^{-15}\) & \(3.55\times 10^{-12}\) & \(6.77\times 10^{-10}\) & \(2.27\times 10^{-7}\) \\ \hline SVD & \(5.55\times 10^{-16}\) & \(1.60\times 10^{-14}\) & \(1.89\times 10^{-13}\) & \(9.09\times 10^{-13}\) \\ \hline DOA & \(1.11\times 10^{-16}\) & \(3.55\times 10^{-15}\) & \(9.09\times 10^{-14}\) & \(7.81\times 10^{-13}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Example 3: ME obtained by the DOA and QR for different \(n\) and a fixed \(q=10\).