Skip to main content

Advertisement

We’d like to understand how you use our websites in order to improve them. Register your interest.

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.

[1234567]

References

  1. 1.

    Aceto L: The Pascal matrix and some numerical methods for ODEs. In Tech. Rep. 01/01. Dipartimento di Energetica, Università degli Studi di Firenze, Florence;

  2. 2.

    Brugnano L, Trigiante : Solving Differential Problems by Multistep Initial and Boundary Value Methods, Stability and Control: Theory, Methods and Applications. Volume 6. Gordon and Breach, Amsterdam; 1998:xvi+418.

  3. 3.

    Dahlquist G: Convergence and stability in the numerical integration of ordinary differential equations. Mathematica Scandinavica 1956, 4: 33–53.

  4. 4.

    Gluchoff A, Hartmann F: Univalent polynomials and non-negative trigonometric sums. The American Mathematical Monthly 1998,105(6):508–522. 10.2307/2589402

  5. 5.

    Henrici P: Applied and Computational Complex Analysis, Pure and Applied Mathematics. Volume 1. John Wiley & Sons, New York; 1974:xv+682.

  6. 6.

    Miller JJH: On the location of zeros of certain classes of polynomials with applications to numerical analysis. Journal of the Institute of Mathematics and Its Applications 1971, 8: 397–406. 10.1093/imamat/8.3.397

  7. 7.

    Mitrinović DS: Analytic Inequalities, Die Grundlehren der mathematischen Wisenschaften. Volume 1965. Springer, New York; 1970:xii+400.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to L Aceto.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Keywords

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