will be negligible and the modified form of equation (16) can be written in a matrix form of AE = t, as-
\begin{equation} \begin{matrix}\left.\ \left[\begin{matrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&0\\ \begin{matrix}\vdots\\ a_{n-1}\\ \end{matrix}&\begin{matrix}\vdots\\ b_{n-1}\\ \end{matrix}&\begin{matrix}\vdots\\ 0\\ \end{matrix}\\ \end{matrix}\ \begin{matrix}0&\ldots&0\\ c_{2}&\ldots&0\\ \begin{matrix}\vdots\\ 0\\ \end{matrix}&\begin{matrix}\ldots\\ \ldots\\ \end{matrix}&\begin{matrix}0\\ c_{n-1}\\ \end{matrix}\\ \end{matrix}\right]\begin{bmatrix}x\\ \begin{matrix}y\\ e_{1}\\ \begin{matrix}\begin{matrix}e_{2}\\ \vdots\\ \end{matrix}\\ e_{n-1}\\ \end{matrix}\\ \end{matrix}\\ \end{bmatrix}=\begin{bmatrix}t_{1}\\ t_{2}\\ \begin{matrix}\vdots\\ t_{n-1}\\ \end{matrix}\\ \end{bmatrix}\right\}&(17)\\ \end{matrix}\nonumber \\ \end{equation}
where\(a_{i}=\frac{2\left(C_{x}-x_{i}\right)}{\text{BL}^{2}},\ b_{i}=\frac{2\left(C_{y}-y_{i}\right)}{\text{BL}^{2}},\ c_{i}=\frac{-2\left(D_{i}\right)}{\text{BL}^{2}},\ and\ t_{i}=\frac{\left(C_{x}^{2}+C_{y}^{2}\right)-\left(x_{i}^{2}+y_{i}^{2}\right)+D_{i}^{2}}{\text{BL}^{2}}\ ;\ \forall i\in(1,\ 2,\ \ldots,\ n-1)\).
Here our objective is to minimize the error, so we solve the equation (17) in such a way that net error must be tended to zero i.e.-
\(\sum_{i=1}^{n-1}{e_{i}\approx 0}\) (18)
Although the net error can’t be zero; but reduce it to close to zero or as minimum as possible.
So the equation (18) can be written by equation (19)-
\begin{equation} \begin{matrix}\text{minimize}\left(\begin{bmatrix}0&0&\overset{\left(n-1\right)\text{times}}{\overbrace{\begin{matrix}1&\begin{matrix}1&\begin{matrix}\ldots&1\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}}}\\ \end{bmatrix}\ \begin{bmatrix}x\\ \begin{matrix}y\\ e_{1}\\ \begin{matrix}\begin{matrix}e_{2}\\ \vdots\\ \end{matrix}\\ e_{n-1}\\ \end{matrix}\\ \end{matrix}\\ \end{bmatrix}\right)&(19)\\ \end{matrix}\nonumber \\ \end{equation}
The equation (19) can be rewritten as (\(\operatorname{}{O^{\prime}E}\)) such that the equation (17) must be true, where \(O^{{}^{\prime}}=\par \begin{bmatrix}0&0&\par \begin{matrix}1&\par \begin{matrix}1&\par \begin{matrix}\ldots&1\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{bmatrix}\), and E’ = \(\par \begin{bmatrix}x&y&\par \begin{matrix}e_{1}&\par \begin{matrix}e_{2}&\par \begin{matrix}\ldots&e_{n-1}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{bmatrix}\).
To find out the minimum value of the summation of errors as shown by equation (18) and (19), some bounded values for the matrix \({}^{\prime}E^{{}^{\prime}}\)are also required. Let ‘lb’ and ‘ub’ are the lower and upper bounds respectively such as-
\(\text{lb}^{{}^{\prime}}=\par \begin{bmatrix}0&0&\par \begin{matrix}0&\par \begin{matrix}0&\par \begin{matrix}\ldots&0\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{bmatrix}_{1\times(2+\left(n-1\right))}\) , and\(\text{ub}^{{}^{\prime}}=\par \begin{bmatrix}\text{BL}&\text{BL}&\par \begin{matrix}2Rc&\par \begin{matrix}2Rc&\par \begin{matrix}\ldots&2Rc\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{bmatrix}_{1\times(2+\left(n-1\right))}\). In a summarized form the objective and its constraints can be defined as-
\(\operatorname{EsuchthatAE=tandlb\leq E\leq ub}{(20)}\)
The solution of the equation (20) leads to an estimation of the unknown node’s coordinates (x, y) as-
\(x=E\left(1\right),\ and\ y=E(2)\).