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On the Ubiquity of the Wrapping Effect in the Computation of Error Bounds

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Perspectives on Enclosure Methods

Abstract

Historically, the wrapping effect was discovered and named in the context of solving ordinary initial value problems in interval arithmetic. Its explanation was obviously geometric: rotations of interval vectors enclosing the set of solutions catch excessive points into the enclosure which may eventually ‘explode’ exponentially. Also discrete dynamical systems share this undesirable behaviour. In the literature the wrapping effect has been discussed primarily in this context.

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References

  1. Alefeld, G., Herzberger, J.: An Introduction to Interval Computations. Academic Press, New York, 1983.

    Google Scholar 

  2. Anguelov, R., Markov, S.: Wrapping effect and wrapping function. Reliab. Comput. 4, No.4, 311–330 (1998).

    Google Scholar 

  3. Berz, M.: Verified integration 01 ODEs and fiows using differential algebraic methods on high-order Taylor models. Reliab. Comput. 4, No.4, 361–369 (1998).

    Google Scholar 

  4. Eijgenraam, P.: The Solution 01 Initial Value Problems Using Interval Arithmetic. Math. Centre Tracts 144, Mathematisch Centrum, Amsterdam (1981).

    Google Scholar 

  5. Gambill, T.N. Skeel, R.D.: Logarithmic Reduction 01 the Wrapping Effect with Applications to Ordinary Differential Equations. SIAM J. Numer. Anal. 25, 153–162 (1988).

    Article  MathSciNet  Google Scholar 

  6. Jackson, L.W.: Interval Arithmetic Error-Bounding Algorithms. SIAM J. Numer. Anal. 12, 223–238 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  7. Kühn, W.: Rigorously Computed Orbits 01 Dynamical Systems Without the Wrapping Effect. Computing 61, No. 1, 47–67 (1998).

    MathSciNet  Google Scholar 

  8. Kühn, W.: Zonotope Dynamics in Numerical Quality Control. In: Hege, H-Ch. (ed.) et al.: Mathematical visualization. Algorithms, applications, and numerics. International workshop Visualization and mathematics, Berlin, Germany, September 16-19, 1997. Berlin: Springer. 125-134 (1998).

    Google Scholar 

  9. Kühn, W.: Towards an Optimal Control of the Wrapping Effect. In: Csendes, Tibor (ed.), Developments in reliable computing. SCAN-98 conference, 8th international symposium on Scientific computing, computer arithmetic and validated numerics. Budapest, Hungary, September 22-25, 1998. Dordrecht: Kluwer Academic Publishers. 43-51 (1999).

    Google Scholar 

  10. Lohner, R.J.: Enclosing the Solutions of Ordinary Initial- and Boundary- Value Problems. In: Kaucher, E., Kuliseh, U., Ullrich Ch. (eds.): Computerarithmetic, pp. 225–286, Teubner Stuttgart (1987).

    Google Scholar 

  11. Lohner, R.J.: Einschlie-ung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen. Dissertation, University of Karlsruhe (1988).

    Google Scholar 

  12. Lohner, R.J.: Verified Computing and Programs in Pascal-XSC. Habilitationsschrift, University of Karlsruhe (1994).

    Google Scholar 

  13. Makino, K, Berz, M.: Efficient control of the dependency problem based on Taylor model methods. Reliab. Comput. 5, No. 1, 3–12 (1999).

    Article  MathSciNet  Google Scholar 

  14. Moore, R.E.: Automatie local coordinate transformations to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations. Error in Digital Comput. 2, Proe. Symp. Madison 1965, 103–140 (1965).

    Google Scholar 

  15. Moore, R.E.: Interval Analysis. Englewood Cliffs, N.J. Prentice-Hall (1966).

    Google Scholar 

  16. Nedialkov, N.S., Jackson, KR., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105, No. 1, 21–68 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  17. Nedialkov, N.S., Jackson, KR.: A New Perspective on the Wrapping Effect in Interval Methods for Initial Value Problems for Ordinary Differential Equations. This Volume. (2001)

    Google Scholar 

  18. Neumaier, A.: Interval methods for systems of equations. Encyclopedia of Mathematics and its Applications, 37. Cambridge etc.: Cambridge University Press. xvi, (1990).

    Google Scholar 

  19. Neumaier, A.: The Wrapping Effect, Ellipsoidal Arithmetic, Stability and Confidence Regions, Computing, Suppl. 9, 175–190 (1993).

    Article  Google Scholar 

  20. Nickel, K: How to fight the wrapping effect. Lect. Notes Comput. Sei. 212, 121–132 (1986).

    Article  Google Scholar 

  21. Rall, L.B.: Automatie Differentiation: Techniques and Applications. Lecture Notes in Computer Science, No. 120, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  22. Stewart, N.P.: A heuristic to reduce the wrapping effect in the numerical solution ofx’ = f(t, x). BIT 11, 328–337 (1971).

    MATH  Google Scholar 

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Lohner, R.J. (2001). On the Ubiquity of the Wrapping Effect in the Computation of Error Bounds. In: Kulisch, U., Lohner, R., Facius, A. (eds) Perspectives on Enclosure Methods. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6282-8_12

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  • DOI: https://doi.org/10.1007/978-3-7091-6282-8_12

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83590-6

  • Online ISBN: 978-3-7091-6282-8

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