Solutions of Overdetermined Systems

  • Joseph L. Awange
  • Béla Paláncz


In geodesy and geoinformatics, field observations are normally collected with the aim of estimating parameters. Very frequently, one has to handle overdetermined systems of nonlinear equations. In such cases, there exist more equations than unknowns, therefore “the solution” of the system can be interpreted only in a certain error metric, i.e., least squares sense.


Unknown Station Overdetermined System Dispersion Matrix Algebraic Technique Combinatorial Solution 
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  1. 68.
    Bähr HG (1988) A quadratic approach to the non-linear treatment of non-redundant observations. Manuscripta Geodaetica 13:191–197Google Scholar
  2. 90.
    Blaha G, Besette RP (1989) Nonlinear least squares method via an isomorphic geometrical setup. Bulletin Geodesique 63:115–138CrossRefGoogle Scholar
  3. 102.
    Brunner FK (1979) On the analysis of geodetic networks for the determination of the incremental strain tensor. Surv Rev 25:146–162CrossRefGoogle Scholar
  4. 176.
    Foulds LR (1984) Combinatorial optimization for undergraduates. Springer, New YorkCrossRefGoogle Scholar
  5. 202.
    Grafarend EW (1985) Variance-covariance component estimation; theoretical results and geodetic applications. Stat Decis 4(Supplement Issue No. 2) 4:407–441Google Scholar
  6. 218.
    Grafarend EW, Sanso F (1985) Optimization and design of geodetic networks. Springer, Berlin/Heidelberg/New York/TokyoCrossRefGoogle Scholar
  7. 219.
    Grafarend EW, Schaffrin B (1974) Unbiased Freenet adjustment. Surv Rev 22:200–218CrossRefGoogle Scholar
  8. 220.
    Grafarend EW, Schaffrin B (1989) The geometry of nonlinear adjustment-the planar trisection problem-. In: Kejlso E, Poder K, Tscherning CC (eds) Festschrift to T. Krarup. Geodaetisk Institut, Copenhagen, 58:149–172Google Scholar
  9. 221.
    Grafarend EW, Schaffrin B (1991) The planar trisection problem and the impact of curvature on non-linear least-squares estimation. Comput Stat Data Anal 12:187–199CrossRefGoogle Scholar
  10. 222.
    Grafarend EW, Schaffrin B (1993) Ausgleichungsrechnung in Linearen Modellen. B. I. Wissenschaftsverlag, MannheimGoogle Scholar
  11. 241.
    Gullikson M, Söderkvist I (1995) Surface fitting and parameter estimation with nonlinear least squares. Zeitschrift für Vermessungswesen 25:611–636Google Scholar
  12. 243.
    Guolin L (2000) Nonlinear curvature measures of strength and nonlinear diagnosis. Allgemein Vermessungs-Nachrichten 107:109–111Google Scholar
  13. 278.
    Hornoch AT (1950) Über die Zurückführung der Methode der kleinsten Quadrate auf das Prinzip des arithmetischen Mittels. Zeitschrift für Vermessungswesen 38:13–18Google Scholar
  14. 285.
    Jacobi CGI (1841) Deformatione et proprietatibus determinantum, Crelle’s Journal für die reine und angewandte Mathematik, Bd. 22Google Scholar
  15. 303.
    Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, Berlin/HeidelbergCrossRefGoogle Scholar
  16. 311.
    Krarup T (1982) Nonlinear adjustment and curvature. In: Forty years of thought, Delft, pp 145–159Google Scholar
  17. 314.
    Kubik KK (1967) Iterative Methoden zur Lösunge des nichtlinearen Ausgleichungsproblemes. Zeitschrift für Vermessungswesen 91:145–159Google Scholar
  18. 365.
    Mautz R (2001) Zur Lösung nichtlinearer Ausgleichungsprobleme bei der Bestimmung von Frequenzen in Zeitreihen. DGK, Reihe C, Nr. 532Google Scholar
  19. 367.
    Meissl P (1982) Least squares adjustment. A modern approach, Mitteilungen der geodätischen Institut der Technischen Universität Craz, Folge 43. 17Google Scholar
  20. 371.
    Mittermayer E (1972) A generalization of least squares adjustment of free networks. Bull Geod 104:139–155CrossRefGoogle Scholar
  21. 381.
    Nicholson WK (1999) Introduction to abstract algebra, 2nd edn. Wiley, New York/Chichester/Weinheim/Brisbane/SingaporeGoogle Scholar
  22. 398.
    Paláncz B, Lewis RH, Zaletnyik P, Awange JL (2008) Computational study of the 3D affine transformation, Part I. 3-point problem, Wolfram Library Archive, MathSource. Accessed 27 Aug 2008
  23. 399.
    Paláncz B, Zaletnyik P, Lewis RH, Awange JL (2008) Computational study of the 3D affine transformation, Part II. N-point problem, Wolfram Library Archive, MathSource. Accessed 27 Aug 2008
  24. 408.
    Perelmuter A (1979) Adjustment of free networks. Bull Geod 53:291–295CrossRefGoogle Scholar
  25. 415.
    Pope A (1982) Two approaches to non-linear least squares adjustments. Can Surv 28:663–669Google Scholar
  26. 417.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in Fortran 77: the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge/New YorkGoogle Scholar
  27. 422.
    Rao CR (1967) Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In: Procedure of the Fifth Barkeley Symposium, BarkeleyGoogle Scholar
  28. 426.
    Rao CR, Kleffe J (1979) Variance and covariance components estimation and applications. Technical report No. 181, Ohio State University, Department of Statistics, ColumbusGoogle Scholar
  29. 454.
    Saito T (1973) The non-linear least squares of condition equations. Bull Geod 110:367–395CrossRefGoogle Scholar
  30. 457.
    Schaffrin B (1983) Varianz-Kovarianz-Komponenten-Schätzung bei der Ausgleichung heterogener Wiederholungsmessungen, DGK, Reihe C, Heft Nr.282Google Scholar
  31. 458.
    Schek HJ, Maier P (1976) Nichtlineare Normalgleichungen zur Bestimmung der Unbekannten und deren Kovarianzmatrix. Zeitschrift für Vermessungswesen 101:140–159Google Scholar
  32. 482.
    Teunissen PJG (1990) Nonlinear least squares. Manuscripta Geodaetica 15:137–150Google Scholar
  33. 484.
    Teunissen PJG, Knickmeyer EH (1988) Non-linearity and least squares. CISM J ASCGC 42:321–330Google Scholar
  34. 514.
    Wellisch S (1910) Theorie und Praxis der Ausgleichsrechnung. Bd. II: Probleme der AusgleichsrechnungGoogle Scholar
  35. 519.
    Werkmeister P (1920) Über die Genauigkeit trigonometrischer Punktbestimmungen. Zeitschrift für Vermessungswesen 49:401–412, 433–456Google Scholar
  36. 547.
    Zaletnyik P, Völgyesi L, Paláncz B (2008) Modelling local GPS/leveling geoid undulations using support vector machines. Period Polytech Civ Eng 52(1):39–43CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Joseph L. Awange
    • 1
  • Béla Paláncz
    • 2
  1. 1.Curtin UniversityPerthAustralia
  2. 2.Budapest University of Technology and EconomicsBudapestHungary

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