Numerical Algorithms

, Volume 82, Issue 3, pp 749–760 | Cite as

Approximating roots of nonlinear systems by α-dense curves

  • G. GarcíaEmail author
Original Paper


We extend an algorithm due to Khamisov (Math. Notes 98(3/4), 484–491, 2015) to approximate, if any exists, a root of a single variable function. For this goal, using the so called α-dense curves, we transform a system of equations of several variables into a single variable equation. The feasibility and limitations of the proposed method are discussed.


Nonlinear equations Numerical methods α-dense curves 

Mathematics Subject Classification (2010)

65H10 49M99 (Primary) 65H20 30C15 (Secondary) 


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This paper is dedicated to my good friend Marga. Also, the author is grateful to the anonymous referees for their suggestions and corrections to improve the quality of the paper.


  1. 1.
    Abaffy, J., Galánti, A.: An always convergent algorithm for global minimization of univariate Lipschitz functions. Acta Polytechnica Hungarica 10(3), 21–39 (2013)Google Scholar
  2. 2.
    Butz, A.R.: Convergence with Hilbert’s space filling curve. J. Comput. System Sci. 3(2), 128–146 (1969)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Butz, A.R.: Solution of nonlinear equations with space filling curves. J. Math. Anal. Appl. 37(2), 351–383 (1972)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cherruault, Y., Mora, G.: Optimisation Globale. Théorie Des Courbes α-Denses. Económica, Paris (2005)Google Scholar
  5. 5.
    Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin (2004)zbMATHGoogle Scholar
  6. 6.
    Galántai, A.: Always convergent methods for solving nonlinear equations. J. Comput. Appl. Mech. 10(2), 183–208 (2015)zbMATHGoogle Scholar
  7. 7.
    Galántai, A.: Always convergent methods for nonlinear equations of several variables. Numer. Algorithms, August, pp. 1–17 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    García, G.: Interpolation of bounded sequences by α-dense curves. Journal of Interpolation and Approximation in Scientific Computing 2017 (1), 1–8 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    García, G., Mora, G., Redwitz, D.A.: Box-counting dimension computed by α-dense curves. Fractals 25(5), 11 pages (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hlawka, E.: Über eine klasse von Näherungspolygonen zur peanokurve. J. Number Theory 43(1), 93–108 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1996)CrossRefGoogle Scholar
  12. 12.
    Hsu, C.S., Zhu, W.H.: A simplicial mappings method for locating the zeros of a function. Quart Appl. Math. 42, 41–59 (1984)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Khamisov, O.V.: Finding roots of nonlinear equations using the method of concave support functions. Math. Notes 98(3/4), 484–491 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mora, G: The peano curves as limit of α-dense curves. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 9(1), 23–28 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mora, G., Cherruault, Y.: Characterization and generation of α-dense curves. Comput. Math. Appl. 33(9), 83–91 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mora, G., Mira, J.A.: Alpha-dense curves in infinite dimensional spaces. Inter. J. of Pure and App. Mathematics 5(4), 257–266 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Mora, G., Redtwitz, D.A.: Densifiable metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105(1), 71–83 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sacco, W.F., Henderson, N.: Finding all solutions of nonlinear systems using a hybrid metaheuristic with fuzzy clustering means. Appl. Soft Comput. 11(8), 5424–5432 (2011)CrossRefGoogle Scholar
  19. 19.
    Sagan, H.: Space-Filling Curves. Springer, New York (1994)CrossRefGoogle Scholar
  20. 20.
    Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to global optimization exploiting space-filling curves, springer briefs in optimization (2013)CrossRefGoogle Scholar
  21. 21.
    Smiley, M.W., Chun, C.: An algorithm for finding all solutions of a nonlinear system. J. Comput. Appl. Math. 137(2), 293–315 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yamamura, Y., Fujioka, T.: Finding all solutions of nonlinear equations using the dual simplex method. J. Comput. Appl. Math. 152(1–2), 587–595 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ziadi, R., Bencherif-Madani, A., Ellaia, A.: Continuous global optimization through the generation of parametric curves. Appl. Math. Comput. 282(5), 65–83 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Zufiria, P.J., Guttalu, R.: On an application of dynamical systems theory to determine all the zeros of a vector function. J. Math. Anal. Appl. 152(1), 269–295 (1990)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Nacional de Educación a Distancia (UNED)AlicanteSpain

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