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A global convergent derivative-free method for solving a system of non-linear equations

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Abstract

Finding all zeros of a system of \(m \in \mathbb {N}\) real non-linear equations in \(n \in \mathbb {N}\) variables often arises in engineering problems. Using Newtons’ iterative method is one way to solve the problem; however, the convergence order is at most two, it depends on the starting point, there must be as many equations as variables and the function F, which defines the system of nonlinear equations F(x)=0 must be at least continuously differentiable. In other words, finding all zeros under weaker conditions is in general an impossible task. In this paper, we present a global convergent derivative-free method that is capable to calculate all zeros using an appropriate Schauder base. The component functions of F are only assumed to be Lipschitz-continuous. Therefore, our method outperforms the classical counterparts.

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References

  1. Abbasbandy, S., Tan, Y., Liao, S.J.: Newton-homotopy analysis method for nonlinear equations. Appl. Math. Comput. 188(2), 1794–1800 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Alefeld, G.: On the convergence of Hallys‘ method. Am. Math. Mon. 88, 530–536 (1981)

    Article  MATH  Google Scholar 

  3. Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burden, R.L., Faires, J.D.: Numerical Analysis, 8th ed. Thomson Brooks/Cole (2005)

  5. Brown, P.R.: A local convergence theory for combined inexact Newton/finite difference methods. SIAM J. Numer. Anal. 24, 407–434 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. George, K.: Numerical Continuation Methods: An Introduction, 388 Pp QA377.A56 640. Springer, New York (1990)

    Google Scholar 

  7. Broyden, C.G.: A class of of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Broyden, C.G.: Quasi-newton methods and their application to function minimization. Math. Comput. 21, 368–381 (1967)

    Article  MATH  Google Scholar 

  9. Dembo, R.S., Eisenstat, S.T., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dennis, J.E., More, J.: Quasi-newton methods: Motivation and theory. SIAM Rev. 19, 46–84 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deuflhard, P.: Newton methods for nonlinear problems. Springer Verlag, Heidelberg, New York (2004)

    MATH  Google Scholar 

  12. Eisenstat, S.C., Walker, H.F.: Globally convergent inexact Newton methods. SIAM J. Optimization 4, 393–422 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kaya, D., El-Sayed, S.M.: A domains‘ decomposition method applied to systems of nonlinear algebraic equations. Appl. Math. Comput. 154, 487–493 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations, Frontiers in Appl. Math., vol. 16. SIAM Publications, Philadelphia, PA (1995)

  15. Kelley, C.T.: Solving nonlinear equations with Newtons‘ Method. SIAM Publications, Philadelphia, PA (2003)

  16. Floudas, A., Pardalos, P.M., Adjiman, C., Esposito, W., Gumus, Z., Harding, S., Klepeis, J., Mayer, C., Schweiger, C.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic, Dordrecht (1999)

  17. Floudas, A.: Recent advances in global optimization for process synthesis, design, and control: enclosure of all solutions. Comput. Chem. Eng. S, 963–973 (1999)

  18. Golbabai, A., Javidi, M.: Newton-like iterative methods for solving system of non-linear equations. Appl. Math. Comput. 192, 546–551 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Gutierrez, J.M., Hernández, M.A.: A family of Chebyshev-Hally type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997)

    Article  MATH  Google Scholar 

  20. Nedzhibov, G.H.: A family of multi-point iterative methods for solving systems of nonlinear equations. J. Comput. Appl. Math. 222, 244–250 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hansen, P., Jaumard, B., Lu, S.H.: On using estimates of Lipschitz constants in global optimization. JOTA 75(1), 195–200 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Homeier, H.H.H.: A modifed Newton method with cubic convergence: The multivariate case. J. Comput. Appl. Math. 169, 161–169 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kalovics, F., Meszaros, G.: Box valued functions in solving systems of equations and inequalities. Numerical Algorithms 36, 1–12 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kubicek, M., Hoffman, H., Hlavacek, V., Sinkule, J.: Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR. Chem. Eng. Sci. 35, 987–996 (1980)

    Article  Google Scholar 

  25. Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC, Boca Raton (2003)

  26. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 47, 499–513 (2004)

    MathSciNet  MATH  Google Scholar 

  27. Liao, S.J.: Notes on the homotopy analysis method: Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul (2008)

  28. Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–355 (2007)

    Article  MathSciNet  Google Scholar 

  29. Monteiro, R.D.C., Pang, J.-S.: A potential reduction Newton method for constrained equations. SIAM J. Optim. 9, 729–754 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Powell, M.J.D. In: Chui, C.K., Schumaker, L.L., Ward, J.D. (eds.) : General algorithms for discrete nonlinear approximation calculations, pp 187–218. Academic Press, New York (1983)

  31. Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. ASME J. Mech. Des. 124, 642–645 (2002)

    Article  Google Scholar 

  32. Qi, L., Sun, J.: A nonsmooth version of Newtons‘ method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ortega, J.M., Rheinboldt, W.C.: Iterative solutions of nonlinear equations in several variables. Academic Press, New York (1970). Russian translation 1976, Chinese translation 1982, reprinted as Classics in Applied Mathematics, Vol 30, SIAM Publications, Philadelphia, PA

  34. Rheinboldt, W.C.: Methods for solving systems of nonlinear equations. In: Regional Conf. Series in Appl. Math., vol. 70, p. 1978. Siam Publications, Philadelphia, PA (1998)

  35. Sherman, A.H.: On Newton-iterative methods for the solution of systems of nonlinear equations. SIAM J. Numer. Anal. 15(4), 755–771 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shi, Y.: A globalization procedure for solving nonlinear systems of equations. Numerical Algorithms 12, 273–286 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sona, T., Musa, M.: Solving systems of nonlinear equations using a globally convergent optimization algorithm Transaction on Evolutionary Algorithm and Nonlinear Optimization (2012)

  38. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2Nd Ed, vol. 12. Springer, New York (1992)

  39. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)

  40. Vrahatis, M.N.: Solving systems of nonlinear equations using the non-zero value of the topological degree. ACM Trans. Math. Softw. 14, 312–329 (1988)

    Article  MATH  Google Scholar 

  41. Walker, H.F., Watson, L.T.: Least-change secant upyear methods for under determined systems. SIAM J. Numer. Math 27, 1227–1262 (1990)

    Article  MATH  Google Scholar 

  42. Weerakoon, S., Fernando, T.G.I.: A variant of Newtons‘ method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–103 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. Werner, W.: Iterative solution of systems of nonlinear equations based upon quadratic approximations. Comput. Math. Appl. 12A(3), 331–343 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wood, G.R.: The bisection method in higher dimensions. Math. Program. 55, 319–337 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wood, G.R., Zhang, B.P.: Estimation of the Lipschitz constant of a function. J. Glob. Optim. 8(1), 91–103 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wu, X.Y.: A new continuation Newton-like method and its deformation. Appl. Math. Comput. 112(1), 75–78 (2000)

    MathSciNet  MATH  Google Scholar 

  47. Ypma, T.J.: Local convergence of inexact Newton methods. SIAM J. Numer. Anal. 21, 583–590 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Technische Universität München, Wissenschaftszentrum Weihenstephan für Ernährung, Landnutzung und Umwelt, Lehrstuhl für Agrarsystemtechnik, Kohlstatt 9, Am Staudengarten 2, 83362 Traunstein, 85354, Munich, Freising-Weihenstephan, Germany

    Sascha Wörz & Heinz Bernhardt

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  1. Sascha Wörz
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  2. Heinz Bernhardt
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Wörz, S., Bernhardt, H. A global convergent derivative-free method for solving a system of non-linear equations. Numer Algor 76, 109–124 (2017). https://doi.org/10.1007/s11075-016-0246-0

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