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One parameter family of linear difference equations and the stability problem for the numerical solution of ODEs

Abstract

The study of the stability properties of numerical methods leads to considering linear difference equations depending on a complex parameter q. Essentially, the associated characteristic polynomial must have constant type for q -. Usually such request is proved with the help of computers. In this paper, by using the fact that the associated polynomials are solutions of a "Legendre-type" difference equation, a complete analysis is carried out for the class of linear multistep methods having the highest possible order.

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Correspondence to L Aceto.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Aceto, L., Pandolfi, R. & Trigiante, D. One parameter family of linear difference equations and the stability problem for the numerical solution of ODEs. Adv Differ Equ 2006, 019276 (2006). https://doi.org/10.1155/ADE/2006/19276

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation