Linear homotopy solution of nonlinear systems of equations in geodesy |
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Authors: | Béla Paláncz Joseph L Awange Piroska Zaletnyik Robert H Lewis |
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Institution: | 1. Department of Photogrammetry and Geoinformatics, Budapest University of Technology and Economics, Pf. 91, 1521, Budapest, Hungary 2. Western Australian Centre for Geodesy, Department of Spatial Sciences, Division of Science and Engineering, Curtin University of Technology, GPO Box U1987, Perth, WA, 6845, Australia 3. Department of Geodesy and Surveying, Budapest University of Technology and Economics, and Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, Pf. 91, 1521, Budapest, Hungary 4. Department of Mathematics, Fordham University, Bronx, NY, 10458, USA
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Abstract: | 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. |
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