New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange
Kalyanasundaram Madhu1, Arul Elango2, René Jr Landry2
1Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, UAE.
Sensors (Basel, Switzerland)
|October 27, 2020
Summary
A new iterative method significantly speeds up Global Navigation Satellite Signal (GNSS) positioning by converging 33% faster than conventional techniques. This enhanced accuracy and efficiency improve user position calculations in GNSS receivers.
Area of Science:
- Numerical analysis
- Satellite navigation systems
- Geodesy
Background:
- Solving systems of nonlinear equations is crucial for many scientific and engineering applications.
- Global Navigation Satellite Signal (GNSS) receivers require accurate and efficient methods for user positioning.
- Existing iterative methods for GNSS positioning can be computationally intensive, requiring multiple iterations.
Purpose of the Study:
- To develop and evaluate a novel two-step fifth-order and multi-step iterative method for solving nonlinear equations.
- To apply these methods to the pseudorange equations in GNSS for improved user positioning.
- To assess the computational efficiency and accuracy compared to conventional iterative techniques.
Main Methods:
- Derivation of a two-step fifth-order and a multi-step iterative method (r≥1).
- Application of the methods to solve nonlinear pseudorange equations in GNSS, considering 4 to 8 satellites.
- Utilization of a fuzzy logic method for satellite selection to optimize Geometrical Dilution of Precision (GDOP).
Main Results:
- The proposed method converges in three iterations, compared to six for conventional methods, reducing computation time.
- Achieved a convergence speed improvement of 33% with 92% accuracy.
- Demonstrated efficient computation and efficiency indices (CE and IE) for the new model.
Conclusions:
- The new iterative method offers a significant improvement in speed and accuracy for GNSS positioning.
- Reduced iteration count leads to quicker position calculations, enhancing GNSS receiver performance.
- The method is effective for solving nonlinear equations in satellite navigation, particularly for pseudorange calculations.
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