Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Rational Runge-Kutta methods for solving systems of ordinary differential equations

Rationale Runge-Kutta Methoden für systeme gewöhnlicher Differentialgleichungen


Some nonlinear methods for solving single ordinary differiential equations are generalized to solve systems of equations. To perform this, a new vector product, compatible with the Samelson inverse of a vector, is defined. Conditions for a given order are derived.


Einige nichtlineare Methoden zur numerischen Lösung skalarer gewöhnlicher Differentialgleichungen werden für Systeme verallgemeinert. Zu diesem Zweck wird ein mit den Samelson-Inversen eines Vektors kompatibles Vektorprodukt definiert. Die Bedingungen für einer vorgegebene Ordnung werden hergeleitet.

This is a preview of subscription content, log in to check access.


  1. [1]

    Padé, H.: Sur la représentation approchée d'une fonction par des fractions rationelles. Am. Sci. Ecole Normale Supér.9, 1–92 (1892).

  2. [2]

    Merson, R. H.: An operational method for the study of investigration processes, Proceedings of a symposiusm on data processing and Automatic Computing Machines at Weapons Research Establishment (1957), Salisbury, Australia. Paper No. 110.

  3. [3]

    Wynn, P.: Acceleration Techniques for Iterated vector and Matrix Problems. Math. Comp.16, 322 (1962).

  4. [4]

    Butcher, J. C.: Coefficients for the study of Runge-Kutta integration processes. J. Aust. Math. Soc.3, 185–201 (1963).

  5. [5]

    Butcher, J. C.: Implicit Runge-Kutta processes. Math. Comp.18, 50–64 (1964).

  6. [6]

    Scraton, R. E.: Estimation of the truncation error in Runge-Kuta and allied processes. The Computer Journal7, 245–248 (1964).

  7. [7]

    Butcher, J. C.: On the attainable order of Runge-Kutta methods. Math. Comp.19 408–417 (1965).

  8. [8]

    Lambert, J. D., Shaw, B.: On the numerical solution ofy′=f(x, y) by a class of formulae based on rational approximation. Math. Comp.19, 456–462 (1965).

  9. [9]

    Curtis, A. R.: Letter to the Editor. The Computer Journal8, 52 (1965).

  10. [10]

    Gragg, W. B.: The Padé table and its relation to certain algorithms of numerical analysis. SIAM Review14, 1–62 (1972).

  11. [11]

    Lambert, J. D.: The unconvential classes of methods for stiff systems, in: Stiff Differential Equations (Willoughby, R., ed.), pp. 171–186. 1974.

  12. [12]

    Lambert, J. D.: Computational methods in ordinary differential equations. London: J. Wiley 1974.

  13. [13]

    Wambecq, A.: Nonlinear methods in solving ordinary differential equations. J. Comp. Appl. Math.2, 27–33 (1976).

  14. [14]

    Gear, G. W.: Numerical initial valu9e problems in ordinary differential equations, pp. 218–219. Englewood Cliffs, N. J.: Prentice-Hall 1971.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wambecq, A. Rational Runge-Kutta methods for solving systems of ordinary differential equations. Computing 20, 333–342 (1978).

Download citation


  • Differential Equation
  • Ordinary Differential Equation
  • Computational Mathematic
  • Vector Product
  • Nonlinear Method