2.3.3. Complexity Analysis

The main computational complexity of the proposed SBB algorithm is related to the number of branch iterations *N* and the convergence accuracy *ε*. In each iteration, the main computational complexity is related to the update of the bounding process of *F*(**x**). More specifically, when we set the norm of the gradient of the solution to be ∇*F* ≤ *ε*, the upper bounds of the complexity required to solve steps (2) and (4) are *O ε*−<sup>2</sup> and *O* 2*ε*−<sup>2</sup> , respectively [48]. Considering the number of branch iterations *N*, the upper bound of the complexity of the SBB algorithm is *O* (2*N* + 1)*ε*−<sup>2</sup> . The upper bounds of the complexity of the following algorithms are shown in the Table 1.

**Table 1.** Algorithms Computational Complexity Comparison.


The above table shows the upper bound of the computational complexity of different algorithms. Among them, the LM algorithm, the Dogleg algorithm, and the TTR algorithm are all Cauchy-related algorithms or Newton-like algorithms, and the upper bound of their complexity is *O ε*−<sup>2</sup> . The complexity of the NR algorithm is related to the number of iterations and the matrix calculation, where *k* is the number of iterations required, and *m* and *n* represent the dimensions of the estimator and the number of equations, respectively. It can be found that the complexity of the SBB algorithm compared with other algorithms

mainly lies in *N*. Since we have shrunk *D* to *D*s, this makes the number of branches *N* usually small, and we will confirm this in simulation experiments.

#### **3. Results**

The SBB algorithm is used to solve the initialization problem of eLoran pseudorange positioning. Therefore, the evaluation of the algorithm is mainly from two aspects. First, the algorithm should still be able to solve the position correctly when no initial value is available, which means that given a random initial value, the algorithm should be able to solve the position accurately. Secondly, the computational complexity of the algorithm should be at a reasonable level so that it can be implemented in the receiver. Based on the above evaluation criteria, this section is organized as follows: we first set the simulation parameters according to the actual station distribution. Then, the performance of various algorithms in solving the eLoran pseudorange positioning problem is compared. Finally, the reliability of the SBB algorithm was verified through simulation.

#### *3.1. Simulation Parameter Settings*

Assuming that the receiver at point A receives the signals from the four eLoran transmitting stations shown in Table 2, the calibrated pseudorange observations and geodesic distance values between point A and eLoran stations are shown in Table 3, and the atmospheric refractive index *ns* is 1.000315. Where the calibrated pseudorange observations *ρ* are as described by Equation (5), they only include the clock deviation *δt* and the observation error *η* caused by time-varying delay factor. We set the clock error *δt* to be 5 *<sup>μ</sup><sup>s</sup>* and *<sup>η</sup>* follows a normal distribution, that is, *<sup>η</sup>* <sup>∼</sup> <sup>N</sup>(0, 50).


**Table 2.** Transmitting station location and coordinates.


