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Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions

  • Michael N. VrahatisEmail author
Conference paper
  • 11 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11974)

Abstract

Generalizations of the traditional intermediate value theorem are presented. The obtained generalized theorems are particular useful for the existence of solutions of systems of nonlinear equations in several variables as well as for the existence of fixed points of continuous functions. Based on the corresponding criteria for the existence of a solution emanated by the intermediate value theorems, generalized bisection methods for approximating fixed points and zeros of continuous functions are given. These bisection methods require only algebraic signs of the function values and are of major importance for tackling problems with imprecise (not exactly known) information.

Keywords

Bolzano theorem Bolzano-Poincaré-Miranda theorem Intermediate value theorems Existence theorems Bisection methods Fixed points Nonlinear equations 

Notes

Acknowledgment

The author would like to thank the anonymous reviewers for their helpful comments.

References

  1. 1.
    Alefeld, G., Frommer, A., Heindl, G., Mayer, J.: On the existence theorems of Kantorovich, Miranda and Borsuk. Electron. Trans. Numer. Anal. 17, 102–111 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bolzano, B.: Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werten, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Prague (1817)Google Scholar
  3. 3.
    Brouwer, L.E.J.: Über Abbildungen von Mannigfaltigkeiten. Math. Ann. 71, 97–115 (1912)Google Scholar
  4. 4.
    Cauchy, A.-L.: Cours d’Analyse de l’École Royale Polytechnique, Paris (1821). (Reprinted in Oeuvres Completes, series 2, vol. 3)Google Scholar
  5. 5.
    Grapsa, T.N., Vrahatis, M.N.: Dimension reducing methods for systems of nonlinear equations and unconstrained optimization: a review. In: Katsiaris, G.A., Markellos, V.V., Hadjidemetriou, J.D. (eds.) Recent Advances in Mechanics and Related Fields - Volume in Honour of Professor Constantine L. Goudas, pp. 343–353. University of Patras Press, Patras (2003)Google Scholar
  6. 6.
    Heindl, G.: Generalizations of theorems of Rohn and Vrahatis. Reliable Comp. 21, 109–116 (2016)MathSciNetGoogle Scholar
  7. 7.
    Jarník, V.: Bernard Bolzano and the foundations of mathematical analysis. In: Bolzano and the Foundations of Mathematical Analysis, pp. 33–42. Society of Czechoslovak Mathematicians and Physicists, Prague (1981)Google Scholar
  8. 8.
    Kavvadias, D.J., Makri, F.S., Vrahatis, M.N.: Locating and computing arbitrarily distributed zeros. SIAM J. Sci. Comput. 21(3), 954–969 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kavvadias, D.J., Makri, F.S., Vrahatis, M.N.: Efficiently computing many roots of a function. SIAM J. Sci. Comput. 27(1), 93–107 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kearfott, R.B.: A proof of convergence and an error bound for the method of bisection in \(\mathbb{R}^n\). Math. Comp. 32(144), 1147–1153 (1978)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kearfott, R.B.: An efficient degree-computation method for a generalized method of bisection. Numer. Math. 32, 109–127 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Knaster, B., Kuratowski, K., Mazurkiewicz, S.: Ein Beweis des Fixpunkt-satzes für \(n\)-dimensionale Simplexe. Fund. Math. 14, 132–137 (1929)CrossRefGoogle Scholar
  13. 13.
    Miranda, C.: Un’ osservatione su un theorema di Brouwer. Bollettino dell’U.M.I. 3, 5–7 (1940)Google Scholar
  14. 14.
    Mourrain, B., Vrahatis, M.N., Yakoubsohn, J.C.: On the complexity of isolating real roots and computing with certainty the topological degree. J. Complex. 18(2), 612–640 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ortega J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Classics in Applied Mathematics, vol. 30. SIAM, PA (2000)Google Scholar
  16. 16.
    Poincaré, H.: Sur certaines solutions particulières du problème des trois corps. Comptes rendus de l’Académie des Sciences Paris 91, 251–252 (1883)zbMATHGoogle Scholar
  17. 17.
    Poincaré, H.: Sur certaines solutions particulières du problème des trois corps. Bull. Astronomique 1, 63–74 (1884)zbMATHGoogle Scholar
  18. 18.
    Scarf, H.: The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math 15(5), 1328–1343 (1967)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sikorski, K.: Bisection is optimal. Numer. Math. 40, 111–117 (1982)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sikorski, K.: Optimal Solution of Nonlinear Equations. Oxford University Press, New York (2001)zbMATHGoogle Scholar
  21. 21.
    Sperner, E.: Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes. Abh. Math. Sem. Hamburg 6, 265–272 (1928)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Stenger, F.: Computing the topological degree of a mapping in \(\mathbb{R}^n\). Numer. Math. 25, 23–38 (1975)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Vrahatis, M.N.: An error estimation for the method of bisection in \(\mathbb{R}^n\). Bull. Greek Math. Soc. 27, 161–174 (1986)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Vrahatis, M.N.: Solving systems of nonlinear equations using the nonzero value of the topological degree. ACM Trans. Math. Softw. 14, 312–329 (1988)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vrahatis, M.N.: CHABIS: a mathematical software package for locating and evaluating roots of systems of nonlinear equations. ACM Trans. Math. Softw. 14, 330–336 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Vrahatis, M.N.: A variant of Jung’s theorem. Bull. Greek Math. Soc. 29, 1–6 (1988)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Vrahatis, M.N.: A short proof and a generalization of Miranda’s existence theorem. Proc. Amer. Math. Soc. 107, 701–703 (1989)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Vrahatis, M.N.: An efficient method for locating and computing periodic orbits of nonlinear mappings. J. Comput. Phys. 119, 105–119 (1995)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Vrahatis, M.N.: Simplex bisection and Sperner simplices. Bull. Greek Math. Soc. 44, 171–180 (2000)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Vrahatis, M.N.: Generalization of the Bolzano theorem for simplices. Topol. Appl. 202, 40–46 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Vrahatis, M.N.: Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros. Topol. Appl. (2019). Accepted for PublicationGoogle Scholar
  32. 32.
    Vrahatis, M.N., Androulakis, G.S., Manoussakis, G.E.: A new unconstrained optimization method for imprecise function and gradient values. J. Math. Anal. Appl. 197(2), 586–607 (1996)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Vrahatis, M.N., Iordanidis, K.I.: A rapid generalized method of bisection for solving systems of non-linear equations. Numer. Math. 49, 123–138 (1986)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Vrahatis, M.N., Ragos, O., Skiniotis, T., Zafiropoulos, F., Grapsa, T.N.: RFSFNS: a portable package for the numerical determination of the number and the calculation of roots of Bessel functions. Comput. Phys. Commun. 92, 252–266 (1995)CrossRefGoogle Scholar
  35. 35.
    Zottou, D.-N.A., Kavvadias, D.J., Makri, F.S., Vrahatis, M.N.: MANBIS—A C++ mathematical software package for locating and computing efficiently many roots of a function: theoretical issues. ACM Trans. Math. Softw. 44(3), 1–7 (2018). Article no. 35CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece

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