## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(a_{i}^{\sharp}\) & E & **O1** & **E** & **02** & **E** & **O3** & **E** & **04** \\ \hline
1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
0.50 & 0.61 & 0.56 & 0.66 & 0.64 & 0.69 & 0.69 & 1.04 & 1.04 \\
0.50 & 0.61 & 0.55 & 0.66 & 0.64 & 0.69 & 0.69 & 1.04 & 1.04 \\
0.20 & 0.39 & 0.31 & 0.43 & 0.40 & 0.60 & 0.58 & 0.50 & 0.57 \\
0.20 & 0.49 & 0.29 & 0.52 & 0.39 & 0.61 & 0.58 & 0.54 & 0.60 \\
0.10 & 0.57 & 0.26 & 0.61 & 0.32 & 0.90 & 0.44 & 0.57 & 0.36 \\
0.10 & 0.91 & 0.27 & 1.03 & 0.36 & 1.64 & 0.45 & 1.34 & 0.34 \\
0.05 & 0.35 & 0.20 & 0.36 & 0.28 & 0.40 & 0.48 & 0.29 & 0.27 \\ \hline \(\bar{d}\) & 0.17 & 0.06 & 0.21 & 0.12 & 0.21 & 0.13 & 0.21 & 0.15 \\ \(\sum v_{i}\) & 7.0 & 17.4 & 7.0 & 11.0 & 7.0 & 6.3 & 7.0 & 6.3 \\ \(\sum n_{i}\) & 24 & 62 & 24 & 48 & 24 & 22 & 24 & 22 \\ \(\sigma_{R}\) & 0.50 & 0.50 & 0.50 & 0.50 & 0.20 & 0.20 & 0.20 & 0.20 \\ \(\sigma_{N}\) & 0.00 & 0.00 & 0.20 & 0.20 & 0.80 & 0.80 & 0.20 & 0.20 \\ \(\sigma_{V}\) & 0.00 & 0.00 & 0.20 & 0.20 & 0.20 & 0.20 & 0.80 & 0.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Uncertainties associated with \(a_{i}\) for four different assignments of \(\sigma_{i}\), \(i=2,\ldots,n=9\). The columns labelled \(E\) give the uncertainties associated with ‘expert judgement’ design given in Table 4, while the columns labelled ‘O’ give those associated with the optimal designs, given, in Table 5, calculated by the SSQR-GE algorithm. Also shown in the table are the aggregate measure \(\bar{d}=|V_{\mathbf{a}}|^{1/n}\) of uncertainty, the total number \(\sum_{i}n_{i}\) of artefacts, and a measure \(\sum_{i}v_{i}\) of the total nominal values involved in each set of experiments. The last three rows give the values of \(\sigma_{R}\), \(\sigma_{N}\), and \(\sigma_{V}\) used to calculate \(\sigma_{i}\), \(i=2,\ldots,n\). ## Appendix A Numerical Linear Algebra

### Sherman-Morrison Formula

See [23]. For \(n\times n\) full rank matrix \(A\) and \(n\)-vectors \(\mathbf{u}\) and \(\mathbf{v}\)

\[\left(A+\mathbf{u}\mathbf{v}^{\top}\right)^{-1}=A^{-1}-\frac{A^{-1}\mathbf{u}\mathbf{v}^{\top}A ^{-1}}{1+\mathbf{v}^{\top}A^{-1}\mathbf{u}}.\]

If \(\tilde{\mathbf{u}}=A^{-1}\mathbf{u}\) and \(\tilde{\mathbf{v}}=A^{-\top}\mathbf{v}\), then

\[\left(A+\mathbf{u}\mathbf{v}^{\top}\right)^{-1}=A^{-1}-\frac{\tilde{\mathbf{u}}\tilde\mathbf{{v}}^{\top}}{1+\tilde{\mathbf{v}}^{\top}\tilde{\mathbf{u}}}=A^{-1}-\frac{\tilde{\mathbf{u}} \tilde{\mathbf{v}}^{\top}}{1+\tilde{\mathbf{v}}^{\top}\mathbf{u}}.\]

### Determinant of a Rank One Update of a Matrix

See, e.g., [42; 43]. For \(A\) and square matrix and \(n\)-vectors \(\mathbf{u}\) and \(\mathbf{v}\)

\[|A+\mathbf{u}\mathbf{v}^{\top}|=|A|(1+\mathbf{v}^{\top}A^{-1}\mathbf{u}).\]

Writing

\[A+\mathbf{u}\mathbf{v}^{\top}=A(I+(A^{-1}\mathbf{u})\mathbf{v}^{\top}),\]

it is sufficient to consider the case \(A=I\): \(|I+\mathbf{u}\mathbf{v}^{\top}|=1+\mathbf{v}^{\top}\mathbf{u}\). From the matrix equation

\[\left[ \begin{array}{cc}I&\mathbf{0}\\ \mathbf{v}^{\top}&1\end{array} \right]\left[ \begin{array}{cc}I+\mathbf{u}\mathbf{v}^{\top }&\mathbf{u}\\ \mathbf{0}&1\end{array} \right]\left[ \begin{array}{cc}I&\mathbf{0}\\ -\mathbf{v}^{\top}&1\end{array} \right]=\left[ \begin{array}{cc}I&\mathbf{u}\\ \mathbf{0}&1+\mathbf{v}^{\top}\mathbf{u}\end{array} \right]\]

it is seen that the determinant of the matrix on the right-hand side is equal to the determinant of the matrix in the middle of the left-hand side, establishing the result for the case \(A=I\). ## Appendix B Legendre Polynomials

See, e.g., [44]. _Note_: in this section, the parameter vector \(\mathbf{a}=(a_{0},\ldots,a_{n})^{\top}\) is an \(n+1\) vector indices starting from \(0\), not \(1\). The D-optimal points for calibration experiments involving a polynomial response function of degree \(n\) are defined in terms of solutions of the polynomial equation

\[F(x)=0,\quad F(x)=(1-x^{2})L_{n-1}(x)=0,\]

where \(L_{n-1}(x)\) is the derivative of Legendre polynomial of degree \(n-1\).