Journal of Geodesy

, 84:79 | Cite as

Linear homotopy solution of nonlinear systems of equations in geodesy

  • Béla PalánczEmail author
  • Joseph L. Awange
  • Piroska Zaletnyik
  • Robert H. Lewis
Original Article


A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers, and has lower complexity compared to other local methods like Newton–Raphson.


Homotopy Nonlinear systems of equations GPS positioning Resection Affine transformation 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Béla Paláncz
    • 1
    Email author
  • Joseph L. Awange
    • 2
  • Piroska Zaletnyik
    • 3
  • Robert H. Lewis
    • 4
  1. 1.Department of Photogrammetry and GeoinformaticsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Western Australian Centre for Geodesy, Department of Spatial Sciences, Division of Science and EngineeringCurtin University of TechnologyPerthAustralia
  3. 3.Department of Geodesy and SurveyingBudapest University of Technology and Economics, and Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of SciencesBudapestHungary
  4. 4.Department of MathematicsFordham UniversityBronxUSA

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