Advances in Difference Equations
Volume 2006 (2006), Article ID 19276, 14 pages
One parameter family of linear difference equations and the stability problem for the numerical solution of ODEs
1Dipartimento di Matematica Applicata “U. Dini,”, Università di Pisa, Via Diotisalvi 2, Pisa 56126, Italy
2Dipartimento di Matematica “U. Dini,”, Università di Firenze, Viale Morgagni 67/A, Firenze 50134, Italy
3Dipartimento di Energetica “S. Stecco,”, Università di Firenze, Via C. Lombroso 6/17, Firenze 50134, Italy
Received 21 July 2004; Accepted 4 October 2004
Copyright © 2006 L. Aceto et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The study of the stability properties of numerical methods leads
to considering linear difference equations depending on a complex
parameter . Essentially, the associated characteristic
polynomial must have constant type for . 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.