## Computer algebra solution of the GPS N-points problem -- by Bela Palancz, Joseph L. Awange, and Erik W. Grafarend

The technical paper which describes this code is published in:
GPS Solutions, Volume 11, Number 4, November 2007, pages 295-299, " Computer algebra solution of the GPS N-points problem"
by Bela Palancz, Dept. of Photogrammetry and GeoInformatics, Budapest University of Technology and Economics, Budapest, Hungary.
Joseph L. Awange, Curtain University of Technology, Perth, Western Australia, Australia.
Erik W. Grafarend, Dept. of Geodesy and Geoinformatics, Suttgart University, Stuttgart, Germany.

Abstract: A computer algebra solution is applied here to develop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four or more pseudoranges at one epoch (the GPS N-points problem). Using Mathematica 5.2 software, the GPS N-points problem is solved numerically, symbolically, semi-symbolically, and with Gauss-Jacobi, on a work station. For the case of N > 4, two minimization approaches based on residuals and distance norms are evaluated for the direct numerical solution and their computational duration is compared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative direct numerical method. For N = 6, the direct numerical solution is twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared to the minimization of the distance norm. Gauss-Jacobi requires eight times more computation time than the direct numerical solution. It does, however, have the advantage of diagnosing poor satellite geometry and outliers. Besides offering a complete evaluation of these algorithms, we have developed Mathematica 5.2 code (a notebook file) for these algorithms (i.e., Sturmfel's resultant, Dixon's resultants, Groebner basis, reduced Groebner basis and Gauss-Jacobi). These are accessible to any geodesist, geophysicist, or geoinformation scientist via the GPS Toolbox website or the Wolfram Information Center (http://library.wolfram.com/infocenter/MathSource/6629/).

The Mathematica notebook file is called GPS_Npoints1.nb . It can be viewed with the older Mathematica reader, or the newer Mathematica player (both are free).