## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
Table 2 gives the computed geometric mean measure \(\bar{d}(V_{\mathbf{\alpha}})\) given in (4) for evenly spaced calibration points, the arcsine solutions, the SSQR solutions, and the SSQR-GE solutions given in \[\mathbf{f}_{i}=V\mathbf{c}_{i},\quad g_{i}^{2}=\mathbf{c}_{i}^{\top}\mathbf{f}_{i}=\mathbf{c}_{i}^ {\top}V\mathbf{c}_{i}. \tag{10}\]2 Also shown is the number of GE exchanges undertaken to improve the SSQR solutions. The table reflects the improvements provided by the SSQR and SSQR-GE algorithms. The number of GE exchanges is small compared to \(m=2001\). The evenly spaced calibration points are markedly worse than the other solutions for larger \(n\). \begin{table}
\begin{tabular}{c c c c|c c c c} \hline \(\mathbf{x_{0}}\) & \(\mathbf{x_{\text{QR}}}\) & \(\mathbf{x_{\text{GE}}}\) & \(\mathbf{x^{*}}\) & \(\mathbf{x_{0}}\) & \(\mathbf{x_{\text{QR}}}\) & \(\mathbf{x_{\text{GE}}}\) & \(\mathbf{x^{*}}\) \\ \hline \multicolumn{8}{c|}{\(n=4\)} & \multicolumn{8}{c}{\(n=5\)} \\ \hline \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) \\ \(-0.500\) & \(-0.488\) & \(-0.447\) & \(-0.447\) & \(-0.707\) & \(-0.669\) & \(-0.655\) & \(-0.655\) \\ \(0.500\) & \(0.437\) & \(0.447\) & \(0.447\) & \(0.000\) & \(0.006\) & \(0.000\) & \(0.000\) \\ \(1.000\) & \(1.000\) & \(1.000\) & \(1.000\) & \(0.707\) & \(0.686\) & \(0.655\) & \(0.655\) \\  & & & & \(1.000\) & \(1.000\) & \(1.000\) & \(1.000\) & \(0.000\) \\ \hline \multicolumn{8}{c|}{\(n=6\)} & \multicolumn{8}{c}{\(n=7\)} \\ \hline \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) \\ \(-0.809\) & \(-0.786\) & \(-0.765\) & \(-0.765\) & \(-0.866\) & \(-0.845\) & \(-0.830\) & \(-0.830\) \\ \(-0.309\) & \(-0.286\) & \(-0.285\) & \(-0.285\) & \(-0.500\) & \(-0.484\) & \(-0.469\) & \(-0.469\) \\ \(0.309\) & \(0.308\) & \(0.285\) & \(0.285\) & \(0.000\) & \(0.002\) & \(0.000\) & \(0.000\) \\ \(0.809\) & \(0.779\) & \(0.765\) & \(0.765\) & \(0.500\) & \(0.493\) & \(0.469\) & \(0.469\) \\ \(1.000\) & \(1.000\) & \(1.000\) & \(1.000\) & \(0.866\) & \(0.841\) & \(0.830\) & \(0.830\) \\  & & & & \(1.000\) & \(1.000\) & \(1.000\) & \(0.000\) \\ \hline \multicolumn{8}{c|}{\(n=8\)} & \multicolumn{8}{c}{\(n=9\)} \\ \hline \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) & \(-1.000\) \\ \(-0.901\) & \(-0.880\) & \(-0.872\) & \(-0.872\) & \(-0.924\) & \(-0.908\) & \(-0.900\) & \(-0.900\) \\ \(-0.623\) & \(-0.613\) & \(-0.592\) & \(-0.592\) & \(-0.707\) & \(-0.692\) & \(-0.677\) & \(Figure 1 graphs the values of \(t_{q}\), \(q=1,\ldots,100\), and its analytical approximation \(\tilde{t}(q)\) given by (12), determined by the sequential D-optimal Algorithm 4, without repeat measurements, for the cases \(n=4\), upper curve, and \(n=11\), lower curve, starting with the SSQR-GE solution. The values of \(t_{q}\) are batched with a significant jump after every \(n\) additional measurement. Without the repeat measurement, the algorithm tends to select available points closest to the D-optimal \(n\) points. If the algorithm is run with repeat measurements allowed, the algorithm selects the same set of D-optimal \(n\) points again and again. This behaviour is reflected in the graphs of \(t_{q}\) for this case; see \[\mathbf{f}_{i}=V\mathbf{c}_{i},\quad g_{i}^{2}=\mathbf{c}_{i}^{\top}\mathbf{f}_{i}=\mathbf{c}_{i}^ {\top}V\mathbf{c}_{i}. \tag{10}\]3