Linear homotopy solution of nonlinear systems of equations in geodesy
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 meani...
| Main Authors: | , , , |
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| Format: | Journal Article |
| Published: |
Springer - Verlag
2010
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| Subjects: | |
| Online Access: | http://hdl.handle.net/20.500.11937/29719 |
| _version_ | 1848752880269393920 |
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| author | Palancz, B. Awange, Joseph Zaletnyik, P. Lewis, R. |
| author_facet | Palancz, B. Awange, Joseph Zaletnyik, P. Lewis, R. |
| author_sort | Palancz, B. |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T08:15:39Z |
| format | Journal Article |
| id | curtin-20.500.11937-29719 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T08:15:39Z |
| publishDate | 2010 |
| publisher | Springer - Verlag |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-297192017-09-13T15:55:37Z Linear homotopy solution of nonlinear systems of equations in geodesy Palancz, B. Awange, Joseph Zaletnyik, P. Lewis, R. GPS positioning Nonlinear systems of equations Resection Affine transformation Homotopy 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. 2010 Journal Article http://hdl.handle.net/20.500.11937/29719 10.1007/s00190-009-0346-x Springer - Verlag restricted |
| spellingShingle | GPS positioning Nonlinear systems of equations Resection Affine transformation Homotopy Palancz, B. Awange, Joseph Zaletnyik, P. Lewis, R. Linear homotopy solution of nonlinear systems of equations in geodesy |
| title | Linear homotopy solution of nonlinear systems of equations in geodesy |
| title_full | Linear homotopy solution of nonlinear systems of equations in geodesy |
| title_fullStr | Linear homotopy solution of nonlinear systems of equations in geodesy |
| title_full_unstemmed | Linear homotopy solution of nonlinear systems of equations in geodesy |
| title_short | Linear homotopy solution of nonlinear systems of equations in geodesy |
| title_sort | linear homotopy solution of nonlinear systems of equations in geodesy |
| topic | GPS positioning Nonlinear systems of equations Resection Affine transformation Homotopy |
| url | http://hdl.handle.net/20.500.11937/29719 |