Linear and Nonlinear Homotopy

Abstract

A fundamental task in geodesy is the solving of 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 the convergence to solutions with no physical meaning, or convergence that requires global method . Although 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 Chapter proposes the Homotopy method that can be implemented easily in high level computer languages like C++ and Fortran, which are faster than the interpreter type CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated by 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 errors, and has a lower complexity compared to other local methods like Newton-Raphson.