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    Linear homotopy solution of nonlinear systems of equations in geodesy

    Access Status
    Fulltext not available
    Authors
    Palancz, B.
    Awange, Joseph
    Zaletnyik, P.
    Lewis, R.
    Date
    2010
    Type
    Journal Article
    
    Metadata
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    Citation
    Palancz, Bela and Awange, Joseph L. and Zaletnyik, Piroska and Lewis, Robert H. 2010. Linear homotopy solution of nonlinear systems of equations in geodesy. Journal of Geodesy. 84 (1): pp. 79-95.
    Source Title
    Journal of Geodesy
    DOI
    10.1007/s00190-009-0346-x
    ISSN
    09497714
    School
    Department of Spatial Sciences
    Remarks

    The original publication is available at: http://www.springerlink.com

    URI
    http://hdl.handle.net/20.500.11937/29719
    Collection
    • Curtin Research Publications
    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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