# New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

#### Literature Survey

## 2. Mathematical Description of New Method and Its Convergence Analysis

#### 2.1. A Two-Step Fifth-Order Method

#### 2.2. A Multi-Step ${(5+3r)}^{th}$-Order Method

#### 2.3. Convergence Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Numerical Examples

**Test Problem 1**(TP1) (see [9])

**Test Problem 2**(TP2) (see [9])

**Test Problem 3**(TP3) (see [9])

## 4. Efficiency of the Methods

## 5. Applications on Global Positioning System (GPS)

#### 5.1. Basics on GPS

#### 5.2. Measurement of Pseudorange

- $c.\Delta t$—Unknown distance caused by the receiver clock offset,
- ${d}_{sat}$ —Advance of the satellite clock with respect to system time,
- ${d}_{iono}$—Ionospheric delay,
- ${d}_{tropo}$—Tropospheric delay,
- ${d}_{rel}$—Relativistic delay, and
- ${d}_{ins}$—Instrumental delay.

#### 5.3. Solving Nonlinear Pseudorange Equations

## 6. Results and Discussion

^{th}order, 8th order and 11th order PM methods almost converged within fourth or fifth iterations with lesser computation time; that is the reason we have chosen only 5

^{th}PM only and later two higher order methods are not included in solving GPS pseudo range equations.

## 7. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Three dimensional user position ([20]).

**Figure 6.**(

**a**) Coordinates used in simulation (Map with skyplot). (

**b**) Coordinates used in simulation (Map with spectrum of GPS signal).

Methods | TP1 | TP2 | TP3 | ||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{M}$ | ${\mathit{err}}_{\mathit{min}}$ | ${\mathit{p}}_{\mathit{c}}$ | $\mathit{M}$ | ${\mathit{err}}_{\mathit{min}}$ | ${\mathit{p}}_{\mathit{c}}$ | $\mathit{M}$ | ${\mathit{err}}_{\mathit{min}}$ | ${\mathit{p}}_{\mathit{c}}$ | |

${2}^{nd}NR$ | 10 | $1.03\times {10}^{-103}$ | 1.99 | 8 | $3.92\times {10}^{-145}$ | 2.00 | 9 | $8.96\times {10}^{-179}$ | 1.99 |

${3}^{rd}TM$ | 7 | $9.65\times {10}^{-104}$ | 2.99 | 6 | $8.87\times {10}^{-236}$ | 3.01 | 6 | $5.79\times {10}^{-142}$ | 2.99 |

${4}^{th}NR$ | 6 | $5.38\times {10}^{-207}$ | 3.99 | 5 | $2.98\times {10}^{-291}$ | 4.03 | 5 | $8.96\times {10}^{-179}$ | 3.99 |

${4}^{th}BCST$ | 6 | $5.05\times {10}^{-139}$ | 3.99 | 5 | $3.49\times {10}^{-238}$ | 4.03 | 5 | $1.61\times {10}^{-142}$ | 3.99 |

${4}^{th}ACT$ | 6 | $2.80\times {10}^{-309}$ | 3.99 | 5 | $3.86\times {10}^{-283}$ | 4.03 | 5 | $2.03\times {10}^{-203}$ | 3.99 |

${4}^{th}SGS$ | 6 | $2.22\times {10}^{-170}$ | 3.99 | 5 | $8.89\times {10}^{-257}$ | 4.03 | 5 | $6.08\times {10}^{-155}$ | 3.99 |

${5}^{th}PM,r=0$ | 5 | $2.41\times {10}^{-114}$ | 4.85 | 4 | $1.46\times {10}^{-131}$ | 5.30 | 5 | $0.00\times {10}^{-0}$ | 4.99 |

${8}^{th}PM,r=1$ | 5 | $0.00\times {10}^{-0}$ | 7.99 | 4 | $0.00\times {10}^{-0}$ | 7.99 | 4 | $1.60\times {10}^{-301}$ | 7.99 |

${11}^{th}PM,r=2$ | 4 | $2.37\times {10}^{-191}$ | 10.96 | 3 | $1.89\times {10}^{-118}$ | 11.65 | 4 | $0.00\times {10}^{-0}$ | 10.99 |

Method | EI | CE |
---|---|---|

${2}^{nd}NR$ | ${2}^{\frac{1}{n+{n}^{2}}}$ | ${2}^{\frac{1}{\frac{1}{3}{n}^{3}+2{n}^{2}+\frac{2}{3}n}}$ |

${3}^{rd}TM$ | ${3}^{\frac{1}{2n+{n}^{2}}}$ | ${3}^{\frac{1}{\frac{1}{3}{n}^{3}+3{n}^{2}+\frac{5}{3}n}}$ |

${4}^{th}NR$ | ${4}^{\frac{1}{2n+2{n}^{2}}}$ | ${4}^{\frac{1}{\frac{2}{3}{n}^{3}+4{n}^{2}+\frac{4}{3}n}}$ |

${4}^{th}BCST$ | ${4}^{\frac{1}{n+2{n}^{2}}}$ | ${4}^{\frac{1}{\frac{2}{3}{n}^{3}+5{n}^{2}+\frac{1}{3}n}}$ |

${4}^{th}ACT$ | ${4}^{\frac{1}{2n+2{n}^{2}}}$ | ${4}^{\frac{1}{\frac{2}{3}{n}^{3}+5{n}^{2}+\frac{4}{3}n}}$ |

${4}^{th}SGS$ | ${4}^{\frac{1}{n+2{n}^{2}}}$ | ${4}^{\frac{1}{\frac{2}{3}{n}^{3}+5{n}^{2}+\frac{4}{3}n}}$ |

${5}^{th}PM$ | ${5}^{\frac{1}{2n+2{n}^{2}}}$ | ${5}^{\frac{1}{\frac{2}{3}{n}^{3}+5{n}^{2}+\frac{4}{3}n}}$ |

${8}^{th}PM$ | ${8}^{\frac{1}{3n+2{n}^{2}}}$ | ${8}^{\frac{1}{\frac{2}{3}{n}^{3}+6{n}^{2}+\frac{7}{3}n}}$ |

${11}^{th}PM$ | ${11}^{\frac{1}{4n+2{n}^{2}}}$ | ${11}^{\frac{1}{\frac{2}{3}{n}^{3}+7{n}^{2}+\frac{10}{3}n}}$ |

X | Y | Z | $\rho $ |
---|---|---|---|

5,731,058.70224 | 14,489,833.1300 | 21,212,629.60495 | 20,626,320.6953 |

14,647,763.94672 | 6,275,922.21243 | 21,269,863.31368 | 22,047,954.0019 |

2,505,594.09552 | 20,672,767.43360 | 16,516,258.99994 | 21,238,160.1101 |

21,359,664.8576 | 12,371,174.84498 | 10,076,109.65011 | 252,188.013893 |

15,520,711.16562 | 1,047,695.53474 | 21,458,920.6173 | 23,317,558.30921 |

12,374,316.48857 | 23,533,071.9923 | 1,008,701.68425 | 23,125,967.55620 |

14,253,263.56829 | 4,103,289.05642 | 21,789,614.16879 | 23,747,495.1073 |

15,759,895.08320 | 5,849,736.59321 | 20,805,729.39674 | 24,788,995.2855 |

Method | M | $\parallel {\mathit{x}}^{(\mathit{k}-1)}-{\mathit{x}}^{(\mathit{k}-2)}{\parallel}_{2}$ | $\parallel {\mathit{x}}^{\left(\mathit{k}\right)}-{\mathit{x}}^{(\mathit{k}-1)}{\parallel}_{2}$ | $\parallel {\mathit{x}}^{(\mathit{k}+1)}-{\mathit{x}}^{\left(\mathit{k}\right)}{\parallel}_{2}$ | CPU(sec) | ${\mathit{p}}_{\mathit{c}}$ | GDOP |
---|---|---|---|---|---|---|---|

${2}^{nd}NR$ | 6 | $0.2271\times {10}^{0}$ | $1.9087\times {10}^{-9}$ | $1.3480\times {10}^{-25}$ | 0.695 | 2.00 | |

${3}^{rd}TM$ | 4 | 2.6206 $\times {10}^{4}$ | $0.0495\times {10}^{0}$ | $3.3212\times {10}^{-19}$ | 0.774 | 3.00 | |

${4}^{th}NR$ | 4 | 2.4777 $\times {10}^{3}$ | 1.9087 $\times {10}^{-9}$ | 2.4095 $\times {10}^{-32}$ | 0.772 | 2.09 | |

${4}^{th}BCST$ | 6 | $21.1858\times {10}^{0}$ | 3.3216$\times {10}^{-5}$ | 8.1651 $\times {10}^{-17}$ | 1.137 | 2.00 | 0.5265 |

${4}^{th}ACT$ | 4 | 1.9374$\times {10}^{3}$ | 7.1420 $\times {10}^{-10}$ | 2.7733 $\times {10}^{-32}$ | 0.903 | 2.06 | |

${4}^{th}SGS$ | 6 | $21.3566\times {10}^{0}$ | 3.3754 $\times {10}^{-5}$ | 8.4317 $\times {10}^{-17}$ | 1.157 | 2.00 | |

${5}^{th}PM$ | 3 | 6.2946 $\times {10}^{6}$ | 3.3716 $\times {10}^{3}$ | 1.6324 $\times {10}^{-12}$ | 0.853 | 4.23 |

Method | M | $\parallel {\mathit{x}}^{(\mathit{k}-1)}-{\mathit{x}}^{(\mathit{k}-2)}{\parallel}_{2}$ | $\parallel {\mathit{x}}^{\left(\mathit{k}\right)}-{\mathit{x}}^{(\mathit{k}-1)}{\parallel}_{2}$ | $\parallel {\mathit{x}}^{(\mathit{k}+1)}-{\mathit{x}}^{\left(\mathit{k}\right)}{\parallel}_{2}$ | CPU(sec) | ${\mathit{p}}_{\mathit{c}}$ | GDOP |
---|---|---|---|---|---|---|---|

${2}^{nd}NR$ | 6 | $0.2577\times {10}^{0}$ | $2.8677\times {10}^{-6}$ | 1.3444$\times {10}^{-11}$ | 0.795 | 2.00 | |

${3}^{rd}TM$ | 5 | $0.3489\times {10}^{0}$ | 2.0554$\times {10}^{-6}$ | 9.6836$\times {10}^{-12}$ | 0.974 | 2.96 | |

${4}^{th}NR$ | 4 | 2.5209$\times {10}^{3}$ | 2.8677 $\times {10}^{-6}$ | $6.3505\times {10}^{-17}$ | 0.838 | 2.27 | |

${4}^{th}BCST$ | 7 | $3.6589\times {10}^{-4}$ | $2.6656\times {10}^{-9}$ | $1.6758\times {10}^{-14}$ | 1.406 | 2.00 | 0.4654 |

${4}^{th}ACT$ | 4 | 1.9676$\times {10}^{3}$ | $1.0730\times {10}^{-7}$ | $2.4238\times {10}^{-18}$ | 1.014 | 2.12 | |

${4}^{th}SGS$ | 7 | $2.8345\times {10}^{-4}$ | $1.6174\times {10}^{-9}$ | $7.6224\times {10}^{-15}$ | 1.460 | 2.00 | |

${5}^{th}PM$ | 3 | $6.3604\times {10}^{6}$ | $2.6147\times {10}^{3}$ | $1.2351\times {10}^{-11}$ | 0.754 | 4.23 |

Method | M | $\parallel {\mathit{x}}^{(\mathit{k}-1)}-{\mathit{x}}^{(\mathit{k}-2)}{\parallel}_{2}$ | $\parallel {\mathit{x}}^{\left(\mathit{k}\right)}-{\mathit{x}}^{(\mathit{k}-1)}{\parallel}_{2}$ | $\parallel {\mathit{x}}^{(\mathit{k}+1)}-{\mathit{x}}^{\left(\mathit{k}\right)}{\parallel}_{2}$ | CPU(sec) | ${\mathit{p}}_{\mathit{c}}$ | GDOP |
---|---|---|---|---|---|---|---|

${2}^{nd}NR$ | 6 | $0.4495\times {10}^{0}$ | $1.4550\times {10}^{-8}$ | $1.4317\times {10}^{-16}$ | 0.866 | 2.00 | |

${3}^{rd}TM$ | 4 | $7.8259\times {10}^{3}$ | $0.0064\times {10}^{0}$ | $5.6447\times {10}^{-11}$ | 0.957 | 2.85 | |

${4}^{th}NR$ | 4 | $2.3459\times {10}^{3}$ | $1.4550\times {10}^{-8}$ | $9.1394\times {10}^{-26}$ | 0.954 | 2.23 | |

${4}^{th}BCST$ | 6 | $54.5307\times {10}^{0}$ | $4.8575\times {10}^{-4}$ | $5.6948\times {10}^{-12}$ | 1.413 | 1.99 | 0.4075 |

${4}^{th}ACT$ | 4 | $2.0707\times {10}^{3}$ | $1.0053\times {10}^{-8}$ | $5.6676\times {10}^{-26}$ | 1.107 | 2.10 | |

${4}^{th}SGS$ | 6 | $54.7063\times {10}^{0}$ | $4.8898\times {10}^{-4}$ | $4.2930\times {10}^{-12}$ | 1.428 | 1.99 | |

${5}^{th}PM$ | 3 | $6.3604\times {10}^{6}$ | $2.5886\times {10}^{3}$ | $3.3911\times {10}^{-11}$ | 0.831 | 4.09 |

**Table 7.**Results of error calculation in 5

^{th}$PM$ and GPS tool box [22] with four satellites scenario.

Iterative Method (GPS Tool Box) | Proposed Method (5^{th}$PM$) | ||
---|---|---|---|

$\mathit{M}$ | Error | $\mathit{M}$ | Error |

1 | $6.3675\times {10}^{6}$ | 1 | $6.2946\times {10}^{6}$ |

2 | $2.0978\times {10}^{5}$ | 2 | $3.3716\times {10}^{3}$ |

3 | $1.2709\times {10}^{3}$ | 3 | $1.6324\times {10}^{-12}$ |

4 | $0.0460$ | ||

5 | $2.8649\times {10}^{-8}$ | ||

6 | $7.7721\times {10}^{-8}$ | ||

7 | 0 |

**Table 8.**Results of error calculation in 5

^{th}$PM$ and GPS tool box [22] with five satellites scenario.

Iterative Method (GPS Tool Box) | Proposed Method (5^{th}$\mathit{PM}$) | ||
---|---|---|---|

$\mathit{M}$ | Error | $\mathit{M}$ | Error |

1 | $1.3363\times {10}^{6}$ | 1 | $6.3604\times {10}^{6}$ |

2 | $6.7220\times {10}^{4}$ | 2 | $2.6147\times {10}^{3}$ |

3 | $168.6654$ | 3 | $1.2351\times {10}^{-11}$ |

4 | 0.0011 | ||

5 | $1.7198\times {10}^{-8}$ | ||

6 | $1.4639\times {10}^{-8}$ | ||

7 | $1.1727\times {10}^{-8}$ | ||

8 | $1.4233\times {10}^{-8}$ | ||

9 | $2.7819\times {10}^{-8}$ | ||

10 | 0 |

**Table 9.**Results of error calculation in 5

^{th}$PM$ and GPS tool box [22] with six satellites scenario.

Iterative Method (GPS Tool Box) | Proposed Method (5^{th}$\mathit{PM}$) | ||
---|---|---|---|

$\mathit{M}$ | Error | $\mathit{M}$ | Error |

1 | $1.9700\times {10}^{6}$ | 1 | $6.3604\times {10}^{6}$ |

2 | $1.1879\times {10}^{4}$ | 2 | $2.5886\times {10}^{3}$ |

3 | $441.8016$ | 3 | $3.3911\times {10}^{-11}$ |

4 | 0.0064 | ||

5 | $2.0352\times {10}^{-8}$ | ||

6 | $1.0844\times {10}^{-8}$ | ||

7 | $1.9246\times {10}^{-8}$ | ||

8 | $9.4634\times {10}^{-9}$ | ||

9 | $1.7549\times {10}^{-8}$ | ||

10 | $3.0064\times {10}^{-9}$ | ||

11 | $5.3703\times {10}^{-8}$ | ||

12 | 0 |

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**MDPI and ACS Style**

Madhu, K.; Elango, A.; Jr Landry, R.; Al-arydah, M.
New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations. *Sensors* **2020**, *20*, 5976.
https://doi.org/10.3390/s20215976

**AMA Style**

Madhu K, Elango A, Jr Landry R, Al-arydah M.
New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations. *Sensors*. 2020; 20(21):5976.
https://doi.org/10.3390/s20215976

**Chicago/Turabian Style**

Madhu, Kalyanasundaram, Arul Elango, René Jr Landry, and Mo’tassem Al-arydah.
2020. "New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations" *Sensors* 20, no. 21: 5976.
https://doi.org/10.3390/s20215976