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-
\(x=E\left(1\right),\ and\ y=E(2)\).