*3.2. Reconstruct Linear Equations*

After filtering outliers, the linear equations in Equation (3) are reconstructed by using the remaining *<sup>m</sup>* measurements as: .

$$
\dot{A}\_{(m)}\theta\_0 = \dot{b}\_{(m)}\tag{15}
$$

However, the linear equations are always close to ill-conditioned, because the elements in the coefficient matrix differ by orders of magnitude. Besides, the unreasonable sensor layout can also result in the linearly dependence of certain equations, in turn to the ill condition of linear equations. The ill condition of linear equations indicates a large difference in the final location results, with a minor change in the coefficient matrix. Moreover, an ill-conditioned linear equation system always has a large condition number of the coefficient matrix . *A*:

$$cond(A) = ||\dot{A}|| \cdot ||\dot{A}^{-1}||\tag{16}$$

Besides, location results deviate largely with a minor disturbance in the coefficient. Therefore, a preconditioning method is applied to linear equations to reduce the condition number and give a more accurate closed-form solution, which is named the preconditioned closed-form solution.

The non-singular and diagonal matrix *P* is found, then the solution of Equation (15) is transformed into that of Equation (17) as:

$$
\dot{P}\dot{A}\theta\_o = P\dot{b} \tag{17}
$$

where *P* = *diag*( <sup>1</sup> *s*1 , 1 *s*2 , ··· , <sup>1</sup> *sn* ) and, *si* = max 1≤*j*≤*m* |(*A*)*ij*|, *i* = 1, 2, ··· , *n*.

Finally, a more accurate location result *θ<sup>o</sup>* can be obtained by:

$$\theta\_o = \left[ \left( \dot{P} \dot{A} \right)^T P \dot{A} \right]^{-1} \left( P \dot{A} \right)^T P \dot{b} \tag{18}$$

due to the improved condition number *cond*(*<sup>P</sup>* . *A*) *cond*( . *A*).
