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

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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.

Keywords:
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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.