## 4 Simulation Results
### Variable-Step GD Estimation
They are compared with respect to the localization errors versus the ground truth positions  
Figure 6: Convergence results (**left column**) with the mean and standard deviations of the number of iterations until convergence (**right column**) over 10,000 simulations of the GD using variable-step descent as a function of the damping parameter, for estimation problems (3)–(6), with and without positioning and measurement error models and by varying the number of neighboring nodes from 4 to 10. (**a**) Problem (3) w/o errors; (**b**) Problem (3) w/o errors; (**c**) Problem (4) w/o errors; (**d**) Problem (4) w/o errors; (**e**) Problem (5) w/errors; (**f**) Problem (5) w/errors; (**g**) Problem (6) w/errors; (**h**) Problem (6) w/errors. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{5}{c}{**Localization Results Based on 8 Relative Measurements**} \\ \hline
**Method** & **Convergence Rate** & **Localization Error (m)** & **Number of Iterations** & **Computation Time (ms)** \\ \hline Fixed-step GD/Equation (3) (w/o errors) & **100.0\%** & **0.006 \(\pm\)** 0.004 & \(31.5\pm 15.6\) & **0.066 \(\pm\)** 0.032 \\ \hline Variable-step GD/Equation (3) (w/o errors) & 95.7\% & 0.006 \(\pm\) 0.004 & \(237\pm 178\) & \(1.7\pm 1.3\) \\ \hline Fixed-step GD/Equation (4) (w/o errors) & **100.0\%** & **0.0000 \(\pm\)** 0.0000 & \(2038\pm 1441\) & \(7.1\pm 5.6\) \\ \hline Variable-step GD/Equation (4) (w/o errors) & 90.0\% & **0.0000 \(\pm\)** 0.0000 & \(775\pm 1138\) & \(3.1\pm 4.4\) \\ \hline Fixed-step GD/Equation (5) (w/errors) & **100.0\%** & 0.091 \(\pm\) 0.032 & \(31.7\pm 15.9\) & **0.133 \(\pm\)** 0.065 \\ \hline Variable-step GD/Equation (5) (w/errors) & 95.9\% & **0.09**\(\pm\) 0.032 & \(239\pm 185\) & \(1.7\pm 1.3\) \\ \hline Fixed-step GD/Equation (6) (w/errors) & **100.0\%** & 0.099 \(\pm\) 0.039 & \(2050\pm 1462\) & \(6.6\pm 5.0\) \\ \hline Variable-step GD/Equation (6) (w/errors) & 90.0\% & 0.099 \(\pm\) 0.039 & \(791\pm 1183\) & \(6.8\pm 10.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparative localization results using the different formulations of the estimation problem based on 8 relative measurements, with and without taking into account uncertainty models. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{5}{c}{**Localization Results Based on 6 Relative Measurements**} \\ \hline
**Method** & **Convergence Rate** & **Localization Error (m)** & **Number of Iterations** & **Computation Time (ms)** \\ \hline Fixed-step GD/Equation (3) (w/o errors) & **96.3\%** & 0.014 \(\pm\) 0.017 & \(38.0\pm 31.2\) & **0.107 \(\pm\)** 0.051 \\ \hline Variable-step GD/Equation (3) (w/o errors) & 95.7\% & 0.015 \(\pm\) 0.017 & \(1281\pm 1282\) & 12.7 \(\pm\) 13.5 \\ \hline Fixed-step GD/Equation (4) (w/o errors) & 95.4\% & **0.0001 \(\pm\)** 0.0001 & \(2105\pm 1649\) & \(5.3\pm 5.2\) \\ \hline Variable-step GD / Problem (4) (w/o errors) & 88.0\% & **0.0001 \(\pm\)** 0.0002 & \(1239\pm 1633\) & \(7.0\pm 10.4\) \\ \hline Fixed-step GD/Equation (5) (w/errors) & 95.3\% & **0.059**\(\pm\) 0.029 & \(38.3\pm 33.4\) & **0.116 \(\pm\)** 0.104 \\ \hline Variable-step GD/Equation (5) (w/errors) & 94.2\% & 0.060 \(\pm\) 0.030 & \(1261\pm 1281\) & \(7.9\pm 8.0\) \\ \hline Fixed-step GD/Equation (6) (w/errors) & **95.5\%** & 0.069 \(\pm\) 0.034 & \(2121\pm 1673\) & \(2.5\pm 2.4\) \\ \hline Variable-step GD/Equation (6) (w/errors) & 87.2\% & 0.071 \(\pm\) 0.036 & \(1232\pm 1644\) & \(9.9\pm 13.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparative localization results using the different formulations of the estimation problem based on 6 relative measurements, with and without taking into account uncertainty models.