A new algorithm for solving all the real roots of a nonlinear system of equations in a given feasible region

  • J. MorenoEmail author
  • Miguel A. López
  • R. Martínez
Original Paper


The initiation of iterations and the encounters of all of its solutions are two of the main problems that are derived from iterative methods. These are produced within feasible regions where the problem lies. This paper provides an algorithm to solve both for the general case of nonlinear systems of p unknowns and q equations. Furthermore, some examples of this algorithm implementation are also introduced.


Nonlinear systems Zeros Algorithms 


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Funding information

This work has been partially supported by MINECO grant no. MTM2014-51891-P, Fundación Séneca de la Región de Murcia grant no. 19219/PI/14, and FEDER OP2014-2020 of Castilla-La Mancha (Spain) grant no. GI20173946.


  1. 1.
    Amat, S., Busquier, S.: After notes on Chebyshev’s iterative method. Appl. Math. Nonlinear Sci. 2(1), 1–12 (2017)CrossRefGoogle Scholar
  2. 2.
    Balibrea, F., Guirao, J.L.G., Lampart, M., Llibre, J.: Dynamics of a Lotka-Volterra map. Fund. Math. 191, 265–279 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comp. 19(92), 577–593 (1965)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bruns, D.D., Bailay, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32, 257–264 (1977)CrossRefGoogle Scholar
  5. 5.
    Chun, C., Neta, B.: Some modification of Newton’s method by the method of undetermined coefficients. Comput. Math. Appl. 56(10), 2528–2538 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Da, P.F., Wu, Q.B., Chan, M.H.: Modified newton-NSS method for solving systems of nonlinear equations. Numer. Algor. 77(1), 1–21 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dennid, J.E., More, J.J.: Quasi-newton methods, motivations and theory. SIAM Rev. 19, 46–89 (1977)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eriksson, J., Gulliksson, M.E.: Local results for the Gauss-Nreyon Method on constrained rank-deficient nonlinear least squares. Math. Comp. 73(248), 1865–1883 (2003)CrossRefGoogle Scholar
  9. 9.
    Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Salanova, M.A.: Chebyshev-like methods and quadratic equations. Rev. Anal. Numr. Thor. Approx. 28, 23–25 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Frontini, M., Sormani, E.: Third-order methods for quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput. 149, 771–782 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Galántai, A.: Always convergent methods for nonlinear equations of several variables. Numer. Algor. 78, 625–641 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guirao, J.L.G., Lampart, M.: Transitivity of a Lotka-Volterra map. Discrete Contin. Dyn. Syst.-ser B 9(1), 75–82 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Iliev, A., Kyurchiev, N.: Nonlinear Methods in Numerical Analysis: Selected Topics in Numerical Analysis. Lap Lambert Academic Publishing, Saarbrucken (2010)Google Scholar
  14. 14.
    Maličky̌, P.: Interior periodic points of a Lotka-Volterra map. J. Differ. Equations Appl. 18(4), 553–567 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Martínez, J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124, 97–121 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Moreno, J., Saiz, A.: Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems. Numer. Algor. 58(2), 203–233 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Moreno, J.: An infinite family of one-step iterators for solving nonlinear equations to increase the order of convergence and a new algorithm of global convergence. Comput. Math. Appl. 66, 1418–1436 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nuño, L., Balbastre, J.V., Rodríguez-Mattalia, S., Jódar, L.: An efficient homotopy continuation method for obtaining the fields in electromagnetic problems when using the MEF with curvilinear elements. In: Proceedings 7th International Conference on Finite Elements for Microwave Engineering Antennas, Circuits and Devices. Madrid (2004)Google Scholar
  19. 19.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)zbMATHGoogle Scholar
  20. 20.
    Pérez, R., Rocha, V.L.: Recent applications and numerical implementation of quasi-Newton methods for solving nonlinear systems of equations. Numer. Algor. 35, 261–285 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rostamy, D., Bakhtiari, P.: New efficient multipoint iterative methods for solving nonlinear systems. Appl. Math. Comput. 266, 350–356 (2015)MathSciNetGoogle Scholar
  22. 22.
    Sharma, J.R., Gupta, P.: An efficient fifth order method for solving systems of nonlinear equations. Comput. Math. Appl. 67(3), 591–601 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Swirszcz, G.: On a certain map of the triangle. Fund. Math. 155(1), 45–57 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Zhang, Y., Huang, P.: High-precition time-interval measurement techniques and methods. Progress in Astronomy 24(1), 1–15 (2006)MathSciNetGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Applied MathematicsETSIE, U.P. de ValenciaValenciaSpain
  2. 2.Department of MathematicsUniversidad de Castilla-La ManchaCuencaSpain

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