GPS Solutions

, Volume 11, Issue 4, pp 295–299 | Cite as

Computer algebra solution of the GPS N-points problem

  • Béla PalánczEmail author
  • Joseph L. Awange
  • Erik W. Grafarend
GPS Tool Box


A computer algebra solution is applied here to develop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four or more pseudoranges at one epoch (the GPS N-points problem). Using Mathematica 5.2 software, the GPS N-points problem is solved numerically, symbolically, semi-symbolically, and with Gauss–Jacobi, on a work station. For the case of N > 4, two minimization approaches based on residuals and distance norms are evaluated for the direct numerical solution and their computational duration is compared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative direct numerical method. For N = 6, the direct numerical solution is twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared to the minimization of the distance norm. Gauss–Jacobi requires eight times more computation time than the direct numerical solution. It does, however, have the advantage of diagnosing poor satellite geometry and outliers. Besides offering a complete evaluation of these algorithms, we have developed Mathematica 5.2 code (a notebook file) for these algorithms (i.e., Sturmfel’s resultant, Dixon’s resultants, Groebner basis, reduced Groebner basis and Gauss–Jacobi). These are accessible to any geodesist, geophysicist, or geoinformation scientist via the GPS Toolbox ( website or the Wolfram Information Center (


GPS CAS Positioning Algebra 



The authors would like to thank Dr. Steve Hilla and the anonymous reviewer for their suggestions and corrections which helped in improving the manuscript. JLA thanks Curtin Research Fellowship.


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Béla Paláncz
    • 1
    Email author
  • Joseph L. Awange
    • 2
  • Erik W. Grafarend
    • 3
  1. 1.Department of Photogrammetry and GeoinformaticsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Western Australian Centre for Geodesy & The Institute for Geoscience ResearchCurtin University of TechnologyPerthAustralia
  3. 3.Department of Geodesy and GeoinformaticsStuttgart UniversityStuttgartGermany

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