Abstract
The problem of reducing the wrapping effeet in interval methods for initial value problems for ordinary differential equations has usually been studied from a geometrie point of view. We develop a new perspeetive on this problem by linking the wrapping effeet to the stability of the interval method. Thus, redueing the wrapping effect is related to finding a more stable seheme for advaneing the solution. This allows us to exploit eigenvalue teehniques and to avoid the eomplieated geometrie arguments used previously.
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References
G. Alefeld and J. Herzberger. Introduction to Interval Computations. Academic Press, New York, 1983.
U. M. Ascher and L. R. Petzold. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, 1998.
C. Barbăroşie. Reducing the wrapping effect. Computing, 54(4):347–357, 1995.
M. Berz and K. Makino. Verified integration of ODEs and ßows using differential algebraic methods on high-order Taylor models. Reliable Computing, 4:361–369, 1998.
P. Eijgenraam. The Solution of Initial Value Problems Using Interval Arithmetic. Mathematical Centre Tracts No. 144. Stichting Mathematisch Centrum, Amsterdam, 1981.
J. G. F. Francis. The QR transformation: A unitary analogue to the LR transformation-part 1. The Computer Journal, 4:265–271, 1961/1962.
T. Gambill and R. Skeel. Logarithmic reduction of the wrapping effect with application to ordinary differential equations. SIAM J. Numer. Anal., 25(1):153–162, 1988.
E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations 1. N onstiff Problems. Springer-Verlag, 2nd revised edition, 1991.
L. W. Jackson. A comparison of ellipsoidal and interval arithIIietic error bounds. In Studies in Numerical Analysis, vol. 2, Proc. Fall Meeting of the Society for Industrial and Applied Mathematics, Philadelphia, 1968. SIAM.
L. W. Jackson. Automatie error analysis for the solution of ordinary differential equations. Technical Report 28, Dept. of Computer Science, University of Toronto, 1971.
L. W. Jackson. Interval arithmetic error-bounding algorithms. SIAM J. Numer. Anal., 12(2):223–238, 1975.
W. Kahan. A computable error bound for systems of ordinary differential equations. Abstract in SIAM Review, 8:568–569, 1966.
F. Krückeberg. Ordinary differential equations. In E. Hansen, editor, Topics in Interval Analysis, pages 91–97. Clarendon Press, Oxford, 1969.
W. Kühn. Rigorously computed orbits of dynamical systems without the wrapping effect. Computing, 61(1):47–67, 1998.
R. J. Lohner. Einschlieftung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen. PhD thesis, Universität Karlsruhe, 1988.
R. E. Moore. Interval Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1966.
N. S. Nedialkov. Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation. PhD thesis, Department of Computer Science, University ofToronto, Toronto, Canada, M5S 3G4, February 1999.
N. S. Nedialkov and K. R. Jackson. atAn interval Hermite-Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. Reliable Computing, 5(3):289-310, 1999. Also in T. Csendes, editor, Developments in Reliable Computing, pp. 289-310, Kluwer, Dordrecht, Netherlands, 1999.
N. S. Nedialkov, K. R. Jackson, and G. F. Corliss. Validated solutions ofinitial value problems for ordinary differential equations. Applied Mathematics and Computation, 105(1):21–68, 1999.
N. S. Nedialkov, K. R. Jackson, and J. D. Pryce. An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE, 2000. Accepted for publication in Reliable Computing, 17 pages.
A. Neumaier. Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory. Computing, 52(4):315–336, 1994.
K. Nickel. How to fight the wrapping effect. In K. Nickel, editor, Interval Analysis 1985, Lecture Notes in Computer Science No. 212, pages 121-132. Springer, Berlin, 1985.
R. Rihm. Interval methods for initial value problems in ODEs. In J. Herzberger, editor, Topics in Validated Computations: Proceedings of the IMACS-GAMM International Workshop on Validated Computations, University of Oldenburg, Elsevier Studies in Computational Mathematics, pages 173–207. Elsevier, Amsterdam, New York, 1994.
N. Stewart. A heuristic to reduce the wrapping effect in the numerical solution of x’ = f(t, x). BIT, 11:328–337, 1971.
D. S. Watkins. Understanding the QR algorithm. SIAM Review, 24(4):427–440, October 1982.
J. H. Wilkinson. The Algebraic Eigenvalue Problem. Oxford Science Publications, Oxford, England, 1965
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Nedialkov, N.S., Jackson, K.R. (2001). A New Perspective on the Wrapping Effect in Interval Methods for Initial Value Problems for Ordinary Differential Equations. In: Kulisch, U., Lohner, R., Facius, A. (eds) Perspectives on Enclosure Methods. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6282-8_13
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DOI: https://doi.org/10.1007/978-3-7091-6282-8_13
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