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. Aceto1
, R. Pandolfi2
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
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.