Abstract
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.
Similar content being viewed by others
References
Allgower EL, Georg K (1990) Numerical continuation methods. An introduction. SCM, vol 13. Springer, Heidelberg
Awange JL, Grafarend EW (2003) Closed form solution of the overdetermined nonlinear 7 parameter datum transformation. Allg Vermessungs-Nachr (AVN) 110: 130–148
Awange JL, Grafarend EW (2005) Solving algebraic computational problems in geodesy and geoinformatics. Springer, Berlin
Bates D, Hauenstein J, Sommese A, Wampler C (2008) Software for numerical algebraic geometry: a paradigm and progress towards its implementation. Software for algebraic geometry. IMA Vol. Math. Appl. vol 148. Springer, New York, pp 1–14
Bernstein DN (1975) The number of roots of a system of equations. Funct Anal Appl 9: 183–185
Bernstein DN, Kushnirenko AG, Khovanskii AG (1976) Newton polyhedra. Uspehi Mat Nauk 31: 201–202
Binous H (2007a) Homotopy continuation method to solve a system of nonlinear algebraic equations. Wolfram Library Archive, MathSource, http://library.wolfram.com/infocenter/MathSource/6710/ [Accessed 1 Dec 2008]
Binous H (2007b) Homotopy continuation method to find all roots of a polynomial equation I–II. Wolfram Library Archive, MathSource, http://library.wolfram.com/infocenter/MathSource/6717/ [Accessed 1 Dec 2008]
Blum L, Cucker F, Shub M, Smale S (1998) Complexity and real computation. Springer, Heidelberg
Choi SH, Book NL (1991) Unreachable roots for global homotopy continuation methods. AIChE J 37 7: 1093–1095
Chapra SC, Canale RP (1998) Numerical methods for engineers, with programming and software applications, 3rd edn. McGraw-Hill, Boston
Decarolis F, Mayer R, Santamaria M (2002) Homotopy continuation methods. http://home.uchicago.edu/~fdc/H-topy.pdf [Accessed 1 Dec 2008]
Drexler FJ (1977) Eine methode zur berechnung sämtlicher lö]sungen von polynomgleichungssystemen. Numer Math 29: 45–58
Garcia CB, Zangwill WI (1979) Determining all solutions to certain systems of nonlinear equations. Math Oper Res 4: 1–14
Garcia CB, Zangwill WI (1981) Pathways to solutions, fixed points and equilibria. Prentice Hall, Englewood Cliffs
Gritton KS, Seader JD, Lin WJ (2001) Global homotopy continuation procedures for seeking all roots of a nonlinear equation. Comput Chem Eng 25: 1003–1019
Grunert JA (1871) Das pothenotsche problem in erweiterter gestalt; nebst bemerkungen über seine anwendungen in der geodäsie. Grunert Archiv für Mathematik Physik 1: 238–241
Gunji T, Kim S, Kojima M, Takeda A, Fujisawa K, Mizutani T (2004) “PHoM” a polyhedral homotopy continuation method for polynomial systems. Computing 73(1): 57–77
Hazaveh K, Jeffrey DJ, Reid GJ, Watt SM, Wittkopf AD (2003) An exploration of homotopy solving in Maple. http://www.apmaths.uwo.ca/~djeffrey/Offprints/ascm2003.pdf[Accessed 27 Aug 2008]
Haneberg WC (2004) Computational geosciences with Mathematica. Springer, Berlin
Kapur D, Saxena T, Yang L (1994) Algebraic and geometric reasoning using Dixon resultants. In: ACM ISSAC 94, international symposium on symbolic and algebraic computation, Oxford, England, pp 99–107
Kotsireas IS (2001) Homotopies and polynomial systems solving I: basic principles. ACM SIGSAM Bull 35(1): 19–32
Lee TL, Li TY, Tsai CH (2008) HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83(2–3): 109–133
Lewis RH (2002) Using the Dixon resultant on big problems. In: CBMS Conference, Texas A&M University, http://www.math.tamu.edu/conferences/cbms/abs.html [Accessed 27 Aug 2008]
Lewis RH (2008) Heuristics to accelerate the Dixon resultant. Math Comput Simul 77(4): 400–407
Leykin A, Verschelde J (2004) PHCmaple: a maple interface to the numerical homotopy algorithms in PHCpack, http://www.ima.umn.edu/~leykin/papers/PHCmaple.pdf [Accessed 1 Dec 2008]
Nakos G, Williams R (2002) A fast algorithm implemented in Mathematica provides one-step elimination of a block of unknowns from a system of polynomial equations, Wolfram Library Archive, MathSource, http://library.wolfram.com/infocenter/MathSource/2597/ [Accessed 27 Aug 2008]
Paláncz B (2006) GPS N-points problem. Math Educ Res 11(2): 153–177
Paláncz B, Awange JL, Grafarend EW (2007) Computer algebra solution of the GPS N-points problem, GPS solutions, 11., 4:1080, Springer, Heidelberg, http://www.springerlink.com/content/75rk6171520gxq72/ [Accessed 1 Dec 2008]
Paláncz B (2008) Introduction to linear homotopy. Wolfram Library Archive, MathSource, http://library.wolfram.com/infocenter/MathSource/7119/ [Accessed 27 Aug 2008]
Paláncz B, Zaletnyik P, Awange JL (2008a) Extension of Procrustes algorithm for 3D affine coordinate transformation. Wolfram Library Archive, MathSource, http://library.wolfram.com/infocenter/MathSource/7171/ [Accessed 27 Aug 2008]
Paláncz B, Lewis RH, Zaletnyik P, Awange JL (2008b) Computational study of the 3D affine transformation, part I. 3-point problem, Wolfram Library Archive, MathSource, http://library.wolfram.com/infocenter/MathSource/7090/ [Accessed 27 Aug 2008]
Paláncz B, Zaletnyik P, Lewis RH, Awange JL (2008c) Computational study of the 3D affine transformation, part II. N-point problem, Wolfram Library Archive, MathSource, http://library.wolfram.com/infocenter/MathSource/7121/ [Accessed 27 Aug 2008]
Papp E, Szucs L (2005) Transformation methods of the traditional and satellite based networks (in Hungarian with English abstract). Geomatikai Kozlemenyek VIII, pp 85–92
Sommese AJ, Verschelde J, Wampler CW (2003) Introduction to numerical algebraic geometry. In: Dickenstein A, Ioannis Z (eds) Emiris solving polynomial equations. Foundations, algorithms and applications. Algorithms and computation in aathematics, vol 14. Springer, Heidelberg, pp 301–337
Sommese AJ, Wampler CW (2005) The Numerical solution of systems of polynomials arising in engineering and science. World Scientific, New Jersey
Verschelde J (1999) Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans Math Softw 25(2): 251–276
Verschelde J (2007) Homotopy methods for solving polynomial systems tutorial at ISSAC’ 05, Beijing, China. http://www.math.uic.edu/~jan/tutorial.pdf [Accessed 1 Dec 2008]
Watson LT, Sosonkina M, Melville RC, Morgan AP, Walker HF (1997) Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms. ACM Trans Math Softw 23(4): 514–549
Zaletnyik P (2008) Application of computer algebra and neural networks to solve coordinate transformation problems. PhD. Thesis at Department of Geodesy and Surveying [in Hungarian], Budapest University of Technology and Economics, Hungary
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Paláncz, B., Awange, J.L., Zaletnyik, P. et al. Linear homotopy solution of nonlinear systems of equations in geodesy. J Geod 84, 79–95 (2010). https://doi.org/10.1007/s00190-009-0346-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-009-0346-x