Extended Newton-Raphson Method

Abstract

In Chap. 7, we have seen that overdetermined nonlinear systems are common in geodetic and geoinformatic applications, that is there are frequently more measurements than it is necessary to determine unknown variables, consequently the number of the variables n is less then the number of the equations m. Mathematically, a solution for such systems can exist in a least square sense. There are many techniques to handle such problems, e.g.,:

• Direct minimization of the residual of the system, namely the minimization of the sum of the least square of the errors of the equations as the objective. This can be done by using local methods, like gradient type methods, or by employing global methods, like genetic algorithms.

• Gauss-Jacobi combinatorial solution. Having more independent equations, m, than variables, n, so m > n, the solution – in a least-squares sense – can be achieved by solving the

  $$\displaystyle{\left \{\begin{array}{c} m\\ n\end{array} \right \}}$$

  combinatorial square subsets (n × n) of a set of m equations, and then weighting these solutions properly. The square systems can be solved again via local methods , like Newton-type methods or by applying computer algebra (resultants, Groebner basis) or global numerical methods, like linear homotopy presented in Chap. 6

• Considering the necessary condition of the minimum of the least square error, the overdetermined system can be transformed into a square one via computer algebra (see ALESS in Sect. 7.2). Then, the square system can be solved again by local or global methods. It goes without saying that this technique works for non-polynomial cases as well.

• For the special type of overdetermined systems arising mostly from datum transformation problems, the so called Procrustes algorithm can be used. There exist different types of them, partial, general and extended Procrustes algorithms. These methods are global and practically they need only a few or no iterations.