Linear homotopy solution of nonlinear systems of equations in geodesy
MetadataShow full item record
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.
The original publication is available at: http://www.springerlink.com
Showing items related by title, author, creator and subject.
Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equationDomairry, G.; Nadim, Nima (2008)Two new analytical methods to solve nonlinear heat transfer equations are homotopy perturbation method and homotopy analysis method. Here, homotopy analysis method, which gives us a vast freedom to choose the answer type, ...
Chai, Qinqin (2013)In this thesis, we develop new computational methods for three classes of dynamic optimization problems: (i) A parameter identification problem for a general nonlinear time-delay system; (ii) an optimal control problem ...
Awange, Joseph; Grafarend, E.; Palancz, B.; Zaletnyik, P. (2010)The book presents modern and efficient methods for solving Geodetic and Geoinformatics algebraic problems. Numerous examples are illustrated with Mathematica using the computer algebra techniques of Ring, Polynomials, ...