## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
Given an estimate \(x\) of a solution, an updated solution is given by [PERSON]'s method [17; 45] according to

\[x:=x-F(x)/\dot{F}(x),\quad\dot{F}(x)=-2x\dot{L}_{n-1}+(1-x^{2})\ddot{L}_{n-1}(x)\]

where \(\ddot{L}\) denotes the second derivative of \(L(x)\) with respect to \(x\). Starting estimates for the \(n+1\) solutions are given by the arcsine points:

\[x_{i}=\cos\left(\pi\frac{n-i}{n}\right),\quad i=0,1,\ldots,n.\]

Legendre polynomials \(L_{j}(x)\) are orthogonal polynomial basis functions defined on the interval \([-1,1]\) and can be evaluated using the three-term recurrence relationship, starting with \(L_{0}(x)=1\), \(L_{1}(x)=x\), and for \(j\geq 2\),

\[jL_{j}(x)=(2j-1)xL_{j-1}(x)-(j-1)L_{j-1}(x).\]If \(f(x)\) is a degree \(n\) polynomial given by a sum of Legendre polynomial basis functions

\[f(x,\mathbf{a})=\sum_{j=0}^{n}a_{j}L_{j}(x),\]

then \(\dot{f}(x)\), the derivative of \(f\) with respect to \(x\), is a degree \(n-1\) polynomial and can be written as

\[\dot{f}(x,\dot{\mathbf{a}})=\sum_{j=0}^{n-1}\dot{a}_{j}L_{j}(x),\]

where \(\dot{\mathbf{a}}=\dot{A}(a_{1},\ldots,a_{n})^{\top}\) is defined in terms of the latter \(n\) elements of the \(n+1\) vector \(\mathbf{a}=(a_{0},a_{1},\ldots,a_{n})^{\top}\), and \(\dot{A}\) is the \(n\times n\) matrix constructed from diagonals formed from the sequence \(1,3,5,\ldots,2n-1\). The matrix \(\dot{A}\) for the case \(n=10\) is

\[\dot{A}=\left[ \begin{array}{ccccccccc}1&0&1&0&1&0&1&0&1&0&1&0\\ 0&3&0&3&0&3&0&3&0&3\\ 0&0&5&0&5&0&5&0&5&0\\ 0&0&0&7&0&7&0&7&0&7\\ 0&0&0&0&9&0&9&0&9&0\\ 0&0&0&0&11&0&11&0&11\\ 0&0&0&0&0&0&13&0&13&0\\ 0&0&0&0&0&0&15&0&15\\ 0&0&0&0&0&0&0&17&0\\ 0&0&0&0&0&0&0&0&19\end{array} \right]\]

Similarly, the second derivative \(\ddot{f}(x)\) can be written as

\[\ddot{f}(x,\ddot{\mathbf{a}})=\sum_{j=0}^{n-2}\ddot{a}_{j}L_{j}(x),\]

where \(\ddot{\mathbf{a}}=\ddot{A}(a_{2},\ldots,a_{n})^{\top}\) is defined in terms of the latter \(n-1\) elements of \(\mathbf{a}=(a_{0},a_{1},\ldots,a_{n})^{\top}\), and \(\ddot{A}\) is the \((n-1)\times(n-1)\) matrix with

\[\ddot{A}=\dot{A}(1:n-1,1:n-1)\dot{A}(2:n,2:n).\]

## References

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