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Linear homotopy solution of nonlinear systems of equations in geodesy

Béla Paláncz
Organization: Budapest University of Technology and Economics
Department: Photogrammetry and Geoinformatics
Joseph Awange
Organization: Curtin University of Technology
Department: Spatial Sciences, Division of Resource and Environmental
Piroska Zaletnyik
Organization: Budapest University of Technology and Economics
Department: Department of Geodesy and Surveying
Robert H. Lewis
Organization: Fordham University
Department: Department of Mathematics
Journal / Anthology

Journal of Geodesy
Year: 2010
Volume: 85
Issue: 1
Page range: 79-95

Afundamental 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 thanCAS 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.'