Advances in Difference Equations 
Volume 2006 (2006), Article ID 19276, 14 pages
doi:10.1155/ADE/2006/19276

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

L. Aceto,1 R. Pandolfi,2 and D. Trigiante3

 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

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∈ℂ−. 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.