Evolutionary Method for Nonlinear Systems of Equations

  • Crina Grosan
  • Ajith Abraham
  • Alexander Gelbukh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4293)


We propose a new perspective for solving systems of nonlinear equations by viewing them as a multiobjective optimization problem where every equation represents an objective function whose goal is to minimize the difference between the right- and left-hand side of the corresponding equation of the system. An evolutionary computation technique is suggested to solve the problem obtained by transforming the system into a multiobjective optimization problem. Results obtained are compared with some of the well-established techniques used for solving nonlinear equation systems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brezinski, C.: Projection methods for systems of equations. Elsevier, Amsterdam (1997)MATHGoogle Scholar
  2. 2.
    Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation 19, 577–593 (1965)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region methods. SIAM, Philadelphia (2000)MATHCrossRefGoogle Scholar
  4. 4.
    Cuyt, A., van der Cruyssen, P.: Abstract Pade approximants for the solution of a system of nonlinear equations. Comp. Math. and Appl. 9, 139–149 (1983)Google Scholar
  5. 5.
    Denis, J.E.: On Newton’s Method and Nonlinear Simultaneous Replacements. SIAM Journal of Numerical Analisys 4, 103–108 (1967)CrossRefGoogle Scholar
  6. 6.
    Denis, J.E.: On Newton–like Methods. Numerical Mathematics 11, 324–330 (1968)CrossRefGoogle Scholar
  7. 7.
    Denis, J.E.: On the Convergence of Broyden’s Method for Nonlinear Systems of Equations. Mathematics of Computation 25, 559–567 (1971)CrossRefGoogle Scholar
  8. 8.
    Denis, J.E., Wolkowicz, H.: Least–Change Secant Methods, Sizing, and Shifting. SIAM Journal of Numerical Analisys 30, 1291–1314 (1993)CrossRefGoogle Scholar
  9. 9.
    Denis, J.E., El-Alem, M., Williamson, K.: A Trust-Region Algorithm for Least-Squares Solutions of Nonlinear Systems of Equalities and Inequalities. SIAM Journal on Optimization 9(2), 291–315 (1999)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Effati, S., Nazemi, A.R.: A new methid for solving a system of the nonlinear equations. Applied Mathematics and Computation 168, 877–894 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldberg, D.E.: Genetic algorithms in search, optimization and machine learning. Addison Wesley, Reading (1989)MATHGoogle Scholar
  12. 12.
    Gragg, W., Stewart, G.: A stable variant of the secant method for solving nonlinear equations. SIAM Journal of Numerical Analisys 13, 889–903 (1976)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970)MATHGoogle Scholar
  14. 14.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge (2002)Google Scholar
  15. 15.
    Steuer, R.E.: Multiple Criteria Optimization. Theory, Computation, and Application. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, New York (1986)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Crina Grosan
    • 1
  • Ajith Abraham
    • 2
  • Alexander Gelbukh
    • 3
  1. 1.Department of Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.IITA Professorship Program, School of Computer Science and EngineeringYonsei UniversitySeoulKorea
  3. 3.Centro de Investigación en Computación (CIC)Instituto Politécnico Nacional (IPN)Mexico

Personalised recommendations