Numerische Mathematik

, Volume 8, Issue 1, pp 29–41 | Cite as

Stronger than uniform convergence of multistep difference methods

  • B. Dejon


This paper is concerned with the numerical integration of ordinary differential equations of the orderx. Sufficient conditions and also necessary ones are given for the s-th difference quotient of the approximate solution to approach thes-th derivative of the exact solution fors >0. This requires a more subtle examination of the multiplicities of the characteristic roots of modulus 1.


Differential Equation Exact Solution Ordinary Differential Equation Approximate Solution Mathematical Method 
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    scDahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Roy. Inst. Technol., Stockholm, No. 130 (4959)Google Scholar
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    Henrici, P.: Discrete variable methods in ordinary differential equations. New York-London: J. Wiley & Sons, Inc. 1962.Google Scholar
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    Kantorovich, L. V.: Functional analysis and applied mathematics. Transl. from the Russian [Uspekhi Matematicheskikh Nauk 3, No. 6, 89–185 (4948)] byC. D. Benster, ed. G. E. Forsythe, National Bureau of Standards Report 1509 (1952).Google Scholar
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    Stetter, H. J.: Asymptotic expansions for the error of discretization algorithms for non-linear functional equations. Num. Math.7, 18–31 (1965)Google Scholar

Copyright information

© Springer-Verlag 1966

Authors and Affiliations

  • B. Dejon
    • 1
  1. 1.IBM Zürich Research LaboratoryZürich

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