Verified numerical computation for nonlinear equations

Article

Abstract

After the introduction basic properties of interval arithmetic are discussed and different approaches are repeated by which one can compute verified numerical approximations for a solution of a nonlinear equation.

Key words

fixed point iteration Newton-like methods nonsmooth equation 

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References

  1. [1]
    G. Alefeld, Intervallrechnung über den komplexen Zahlen und einige Anwendungen. Ph.D. thesis, Universität Karlsruhe, Karlsruhe, 1968.Google Scholar
  2. [2]
    G. Alefeld, Inclusion methods for systems of nonlinear equations—the interval Newton method and modifications. Topics in Validated Computations, J. Herzberger (ed.), Elsevier, Amsterdam, 1994, 7–26.Google Scholar
  3. [3]
    G. Alefeld, X. Chen and F. Potra, Numerical validation of solutions of linear complementarity problems. Numer. Math.,83 (1999), 1–23.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    G. Alefeld, X. Chen and F. Potra, Numerical validation of solutions of complementarity problems: the nonlinear case. Numer. Math.,92 (2002), 1–16.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    G. Alefeld and J. Herzberger, Einführung in die Intervallrechnung. Bibliographisches Institut, Mannheim, 1974.MATHGoogle Scholar
  6. [6]
    G. Alefeld and J. Herzberger, Introduction to Interval Computations., Academic Press, New York, 1983.MATHGoogle Scholar
  7. [7]
    G. Alefeld and G. Mayer, Interval analysis. Theory and applications. J. Comp. Appl. Math.,121 (2000), 421–464.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Alefeld and U. Schäfer, Iterative methods for linear complementarity problems with interval data. Computing,70 (2003), 235–259.MATHMathSciNetGoogle Scholar
  9. [9]
    G. Alefeld, Z. Shen and Z. Wang, Enclosing solutions of linear complementarity problems for H-matrices. Reliable Computing,10 (2004), 423–435.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Alefeld and Z. Wang, Verification of solutions of almost linear complementarity problems. Annals of the European Academy of Sciences 2005, EAS Publishing House, 211–231.Google Scholar
  11. [11]
    X. Chen, A verification method for solutions of nonsmooth equations. Computing,58 (1997), 281–294.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing,4 (1969), 187–201.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. Neumaier, Interval Methods for Systems of Equations. University Press, Cambridge, 1990.MATHGoogle Scholar
  14. [14]
    H. Schwandt, Schnelle fast global konvergente Verfahren für die Fünf-Punkte-Diskrestisierung der Poissongleichung mit Dirichletschen Randbedingungen auf Rechteckgebieten. Thesis, Fachbereich Mathematik der TU Berlin, Berlin, 1981.Google Scholar

Copyright information

© JJIAM Publishing Committee 2009

Authors and Affiliations

  1. 1.University of KarlsruheKarlsruheGermany

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