18b is equivalent to interpolating by a function of

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Unformatted text preview: an be computed as the difference between the rst subdiagonal element and the diagonal or by the di erence between two successive diagonal elements. 2. Extrapolation methods can be written as Runge-Kutta methods if the step size and order of the extrapolation are xed. In this case, convergence, stability, and error bounds follow from the results of Chapter 3 for general one-step methods. 3. Implicit methods can be used in combination with extrapolation to solve sti problems. A survey of the state of the art was written by Deu hard 3] who prefers polynomial extrapolation to rational extrapolation because rational extrapolation lacks translation invariance, rational extrapolation can impose restrictions on the base step size, and polynomial extrapolation is slightly more e cient. 8 Bibliography 1] R. Bulirsch and J. Stoer. Fehlerabschatzungen und extrapolation mit rationalen funktionen bei verfahren vom Richardson-typus. Numer. Math., 6:413{427, 1964. 2] R. Bulirsch and J. Stoer. Numerical treatment of ordinary di erential equations by extrapolation methods. Numer. Math., 8:1{13, 1966. 3] P. Deu hard. Recent progress in extrapolation methods for ordinary di erential equations. SIAM Review, 27:505{535, 1985. 4] W.B. Gragg. On extrapolation algorithms for ordinary initial value problems. SIAM J. Numer. Anal., 2:384{403, 1964. 5] E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Di erential Equations I: Nonsti Problems. Springer-Verlag, Berlin, second edition, 1993. 9...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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